Exponents And Scientific Notation - OpenStax CNX
OpenStax-CNX module: m512411Exponents and Scientific Notation*OpenStaxOpenStax Algebra and TrigonometryThis work is produced by OpenStax-CNX and licensed under theCreative Commons Attribution License 4.0 AbstractIn this section students will: Use the product rule of exponents. Use the quotient rule of exponents. Use the power rule of exponents. Use the zero exponent rule of exponents. Use the negative rule of exponents. Find the power of a product and a quotient. Simplify exponential expressions. Use scienti c notation.Mathematicians, scientists, and economists commonly encounter very large and very small numbers. Butit may not be obvious how common such gures are in everyday life. For instance, a pixel is the smallestunit of light that can be perceived and recorded by a digital camera. A particular camera might record animage that is 2,048 pixels by 1,536 pixels, which is a very high resolution picture.It can also perceive acolor depth (gradations in colors) of up to 48 bits per frame, and can shoot the equivalent of 24 frames persecond. The maximum possible number of bits of information used to lm a one-hour (3,600-second) digitallm is then an extremely large number.Using a calculator, we enter 2, 048 1, 536 48 24 3, 600 and press ENTER. The calculator displays1.304596316E13. What does this mean? The E13 portion of the result represents the exponent 13 of ten,so there are a maximum of approximately 1.3 1013 bitsof data in that one-hour lm. In this section, wereview rules of exponents rst and then apply them to calculations involving very large or small numbers.1 Using the Product Rule of ExponentsConsider the product x3· x4 . Bothterms have the same base, x, but they are raised to di erent exponents.Expand each expression, and then rewrite the resulting expression.x3 · x43factors7* Version 1.8: May 12, 2016 9:56 am -0500 cnx.org/content/m51241/1.8/4factors x·x·x·x·x·x·xfactors x·x·x·x·x·x·x x7(1)
OpenStax-CNX module: m51241The result is that x32· x4 x3 4 x7 .Notice that the exponent of the product is the sum of the exponents of the terms. In other words, whenmultiplying exponential expressions with the same base, we write the result with the common base and addthe exponents. This is the product rule of exponents.am · an am n(2)23 · 24 23 4 27(3)Now consider an example with real numbers.34We can always check that this is true by simplifying each exponential expression. We nd that 2 is 8, 2 is716, and 2 is 128. The product 8· 16 equals128, so the relationship is true. We can use the product rule ofexponents to simplify expressions that are a product of two numbers or expressions with the same base butdi erent exponents.A General Note:For any real number a and natural numbers m and n,the product rule ofexponents states thatam · an am n(4)Example 1Using the Product RuleWrite each of the following products with a single base. Do not simplify further.a.b.c.t5 · t35( 3) · ( 3)x2 · x5 · x3SolutionUse the product rule to simplify each expression.a.b.c.t5 · t3 t5 3 t85515 16( 3) · ( 3) ( 3) · ( 3) ( 3) ( 3)253x ·x ·xAt rst, it may appear that we cannot simplify a product of three factors.However, using theassociative property of multiplication, begin by simplifying the rst two. x2 · x5 · x3 x2 · x5 · x3 x2 5 · x3 x7 · x3 x7 3 x10(5)Notice we get the same result by adding the three exponents in one step.x2 · x5 · x3 x2 5 3 x10http://cnx.org/content/m51241/1.8/(6)
OpenStax-CNX module: m512413Try It:Exercise 2(Solution on p. 24.)Write each of the following products with a single base. Do not simplify further.69 · k4 2· y2b.ya.kc.t3· t6 · t52 Using the Quotient Rule of ExponentsThe quotient rule of exponents allows us to simplify an expression that divides two numbers with the samebase but di erent exponents.asymy n ,where m n. ConsiderIn a similar way to the product rule, we can simplify an expression suchthe exampley9y 5 . Perform the division by canceling common factors.y9y5 )yNotice that the exponent of the quotient is the di erence between the exponents of the divisor and dividend.am am nan(8)In other words, when dividing exponential expressions with the same base, we write the result with thecommon base and subtract the exponents.y9 y 9 5 y 4y5For the time being, we must be aware of the condition m n. Otherwise,(9)the di erence m n could be zeroor negative. Those possibilities will be explored shortly. Also, instead of qualifying variables as nonzero eachtime, we will simplify matters and assume from here on that all variables represent nonzero real numbers.A General Note:For any real number a and natural numbers m and n,such that m n,thequotient rule of exponents states thatam am nanExample 2Using the Quotient RuleWrite each of the following products with a single base. Do not simplify further.( 2)14( 2)9t23b. 15t 5(z 2)c.z 2a.http://cnx.org/content/m51241/1.8/(10)
OpenStax-CNX module: m512414SolutionUse the quotient rule to simplify each expression.14 95( 2)14 ( 2) ( 2)( 2)9t2323 15b. 15 t t8t 5 4 5 12z( )c. z 2 z 2z 2a.Try It:Exercise 4(Solution on p. 24.)Write each of the following products with a single base. Do not simplify further.s75s68( 3)6b. 35(ef 2 )c.(ef 2 )3a.3 Using the Power Rule of ExponentsSuppose an exponential expression is raised to some power. Can we simplify the result? Yes. To do this,we use the power rule of exponents. Consider the expression 3x2 . Theexpression inside the parenthesesis multiplied twice because it has an exponent of 2. Then the result is multiplied three times because theentire expression has an exponent of 3.3x2 3factors x2 · x2 · x2 32factors factors 2 factorsz} {z} {z} { x·x · x·x · x·x 2factors x·x·x·x·x·x x6The exponent of the answer is the product of the exponents:x2 3(11) x2·3 x6 . In other words, when raisingan exponential expression to a power, we write the result with the common base and the product of theexponents.n(am ) am·n(12)Be careful to distinguish between uses of the product rule and the power rule. When using the product rule,di erent terms with the same bases are raised to exponents. In this case, you add the exponents. Whenusing the power rule, a term in exponential notation is raised to a power. In this case, you multiply theexponents.http://cnx.org/content/m51241/1.8/
OpenStax-CNX module: m512415Product Rule53 · 545Power Rule 53 4 57but 453 x5 2 x7butx5 (3a) 53·4 2 x5·2 12x5 · x2x10710(3a) · (3a) 7 10(3a)17but 7(3a) 10 7·10(3a) 70(3a)A General Note:For any real number a and positive integers m and n,the power rule of expo-nents states thatn(am ) am·n(14)Example 3Using the Power RuleWrite each of the following products with a single base. Do not simplify further.a.b.c. 7x2 35(2t) 115( 3)SolutionUse the power rule to simplify each expression.a.b.c. 7x2 x2·7 x14 355·315(2t) (2t) (2t) 1155·1155( 3) ( 3) ( 3)Try It:Exercise 6(Solution on p. 24.)Write each of the following products with a single base. Do not simplify further. 38(3y) 5 7b. t 44c. ( g)a.http://cnx.org/content/m51241/1.8/(13)
OpenStax-CNX module: m5124164 Using the Zero Exponent Rule of ExponentsReturn to the quotient rule. We made the condition that mbe zero or negative.What would happen if m n?In n sothat the di erence m n wouldneverthis case, we would use the zero exponent rule ofexponents to simplify the expression to 1. To see how this is done, let us begin with an example.)t8t8 1t8)t8(15)If we were to simplify the original expression using the quotient rule, we would havet8 t8 8 t0t80If we equate the two answers, the result is t 1. This(16)is true for any nonzero real number, or any variablerepresenting a real number.a0 10The sole exception is the expression 0. This(17)appears later in more advanced courses, but for now, we willconsider the value to be unde ned.For any nonzero real number a,the zero exponent rule of exponents statesA General Note:thata0 1Example 4Using the Zero Exponent RuleSimplify each expression using the zero exponent rule of exponents.c3c3 3x5b.x54(j 2 k)c.(j 2 k)·(j 2 k)325(rs2 )d.(rs2 )2a.SolutionUse the zero exponent and other rules to simplify each expression.c3c3a. 3x5x5 c3 3 c0 1 3 · 3 · xb.x5x55 5 3 · x0 3 · 1 3http://cnx.org/content/m51241/1.8/(18)
OpenStax-CNX module: m5124174(j 2 k)(j 2 k)·(j 2 k)34 c. (j 2 k)Use the product rule in the denominator.(j 2 k)1 34(j 2 k)4(j 2 k) 4 42Simplify.j kj2k Use the quotient rule. 0Simplify. 25(rs2 )(rs2 )2d.1 2 2 5 rs2 0 5 rs2Use the quotient rule.Simplify. 5·1Use the zero exponent rule. 5Simplify.Try It:Exercise 8(Solution on p. 24.)Simplify each expression using the zero exponent rule of exponents.t7t711(de2 )b.2(de2 )11w4 ·w2c.w6t3 ·t4d. 2 5t ·ta.5 Using the Negative Rule of ExponentsAnother useful result occurs if we relax the condition that m n in the quotient rule even further. Forh3?Whenm n thatis,wherethedi erencem n is negative we can use theh5negative rule of exponents to simplify the expression to its reciprocal.h3Divide one exponential expression by another with a larger exponent. Use our example, 5 .hexample, can we simplifyh3h5 h1h2(19)If we were to simplify the original expression using the quotient rule, we would haveh3h5 h3 5 2Putting the answers together, we have hrepresenting a nonzero real number.http://cnx.org/content/m51241/1.8/ h 2(20)1h2 . This is true for any nonzero real number, or any variable
OpenStax-CNX module: m512418A factor with a negative exponent becomes the same factor with a positive exponent if it is moved acrossthe fraction bar from numerator to denominator or vice versa.a n 1anandnWe have shown that the exponential expression anof a natural number. That means that aan 1a n(21)is de ned when n is a natural number, 0, or the negativeis de ned for any integer n. Also, the product and quotient rulesand all of the rules we will look at soon hold for any integer n.A General Note:For any nonzero real number a and natural number n,the negative rule ofexponents states thata n 1an(22)Example 5Using the Negative Exponent RuleWrite each of the following quotients with a single base. Do not simplify further. Write answerswith positive exponents.θ3θ 10z 2 ·zb.z44( 5t3 )c.83( 5t )a.Solutionθ33 10 θ 7 θ17θ 10 θ22 1z ·zzz33 4b. z 1 z1z4 z4 z4 z3 4 44 8( 5t )c. 5t3 5t3 ( 5t1 3 )4( 5t3 )8a.Try It:Exercise 10(Solution on p. 24.)Write each of the following quotients with a single base. Do not simplify further. Writeanswers with positive exponents.( 3t)2( 3t)8f 47b. 49f ·f2k4c. 75ka.Example 6Using the Product and Quotient RulesWrite each of the following products with a single base. Do not simplify further. Write answerswith positive exponents.a.b2 · b 8http://cnx.org/content/m51241/1.8/
OpenStax-CNX module: m512419 55( x) · ( x)b. 7zc.( 7z)5Solutiona.b.c.b2 · b 8 b2 8 b 6 b165 55 50( x) · ( x) ( x) ( x) 11 5 4( 7z)1 7z ( 7z)5 ( 7z) ( 7z) ( 7z)51( 7z)4Try It:Exercise 12(Solution on p. 24.)Write each of the following products with a single base. Do not simplify further. Writeanswers with positive exponents.a.t 112512b. 1325· t66 Finding the Power of a ProductTo simplify the power of a product of two exponential expressions, we can use the power of a product rule ofexponents, which breaks up the power of a product of factors into the product of the powers of the factors.3For instance, consider (pq). Webegin by using the associative and commutative properties of multiplicationto regroup the factors.3(pq)3 (pq) · (pq) · (pq)p·q·p·q·p·q3In other words, (pq)3factorsfactors3(23)factors p·p·p · q·q·q p3 · q 3 p3 · q 3 .A General Note:For any real numbers a and b and any integer n,the power of a product rule ofexponents states thatn(ab) an bn(24)Example 7Using the Power of a Product RuleSimplify each of the following products as much as possible using the power of a product rule.Write answers with positive exponents.a.b. 3ab215(2t)http://cnx.org/content/m51241/1.8/
OpenStax-CNX module: m51241c.d.e. 2w310 31( 7z)4 2 2 7efSolutionUse the product and quotient rules and the new de nitions to simplify each expression.a.b.c.d.e. 3 33ab2 (a) · b2 a1·3 · b2·3 a3 b6151515(2t) (2) · (t) 215 t15 32, 768t15 333 2w3 ( 2) · w3 8 · w3·3 8w911 ( 7)41·(z)4 2,401z4( 7z)4 2 2 7 2 72 7e f e· f e 2·7 · f 2·7 e 14 f 14 f 14e14Try It:Exercise 14(Solution on p. 24.)Simplify each of the following products as much as possible using the power of a productrule. Write answers with positive exponents. 5g 2 h33b.(5t) 5 3c. 3ya.d.e.1(a6 b7 )3 3 2 4r s7 Finding the Power of a QuotientTo simplify the power of a quotient of two expressions, we can use the power of a quotient rule, which statesthat the power of a quotient of factors is the quotient of the powers of the factors. For example, let's lookat the following example.e 2 f 2 7 f 14e14(25)Let's rewrite the original problem di erently and look at the result.e 2 f 2 7 7f2e2f 14e14(26)It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule.http://cnx.org/content/m51241/1.8/
OpenStax-CNX module: m5124111e 2 f 2 7 7(f 2 ) 7(e2 )7(27)f 2·7e2·7f 14e14 A General Note:f2e2For any real numbers a and b and any integer n,the power of a quotient ruleof exponents states that a nb anbn(28)Example 8Using the Power of a Quotient RuleSimplify each of the following quotients as much as possible using the power of a quotient rule.Write answers with positive exponents. 4 311z 6pb.q3 1 27c.t2 3 2 4d. j ka.e.m 2 n 2 3Solutiona.b.c.d.e. 364644 3 (z(4)11 )3 z 11·3 z 33z 11 66pp1·6p6 (q(p)3 )6 q 3·6 q 18q3 27 1 27 11 ( 1) t 12·27 t54 t54t2(t2 )27 44 433·412(j 3 )j 3 k 2 kj 2 (k2 )4 kj 2·4 jk8 3 331m 2 n 2 m21n2 (m(1)2 n2 )3 (m2 )3 (n2 )3 1m2·3 ·n2·3 1m6 n6Try It:Exercise 16(Solution on p. 24.)Simplify each of the following quotients as much as possible using the power of a quotientrule. Write answers with positive exponents. 5 3a.bc 5 4u8 1 35c.w3 4 3 8d. pqb.e.c 5 d 3 4http://cnx.org/content/m51241/1.8/
OpenStax-CNX module: m51241128 Simplifying Exponential ExpressionsRecall that to simplify an expression means to rewrite it by combing terms or exponents; in other words, towrite the expression more simply with fewer terms. The rules for exponents may be combined to simplifyexpressions.Example 9Simplifying Exponential ExpressionsSimplify each expression and write the answer with positive exponents only.a.b.c.d.e.f. 36m2 n 1 4 3175 · 17 · 17 2u 1 vv 1 3 1 2a b5a 2 b2 4 2 42x 2 x 25(3w2 ) (6w 2 )2Solution6m2 n 1 3a.3 (6) m2 3 3n 1 32·3 1·36 mn6 3 216m n216m6n35 4 3 175 · 17 4 · 17 3 b.17 2 1711172 or 289 1u vv 1 2 c. 2(u 1 v)v 2 4u 2a3 b5a 2 b 2 · 5 · a3 · a 2 · b 1 · b2 5(3w2 )(6w 2 )2The negative exponent ruleCommutative and associative laws of multiplication 10 · a3 2 · b 1 2 10ab 4 4x 2 0x2 221The product ruleSimplify.The product ruleSimplify.The zero exponent rule5 f.Simplify.Simplify. e.The product ruleThe negative exponent ruled. 4 4x 2 x2 2The negative exponent ruleThe quotient rulev 22Simplify.The power of a product rulev4u2 1The power ruleThe power of a quotient rule(v 1 )2u 2 v 2v 2 2 2 ( 2) uThe power of a product rule (3)5 ·(w2 )(6)2 ·(w 2 )235 w2·562 w 2·2243w1036w 427w10 ( 4)427w144http://cnx.org/content/m51241/1.8/The power of a product ruleThe power ruleSimplify.The quotient rule and reduce fractionSimplify.
OpenStax-CNX module: m5124113Try It:Exercise 18(Solution on p. 24.)Simplify each expression and write the answer with positive exponents only. 32uv 28 12b.x · x ·x a.c.d.e.f.2e2 f 3f 1 5 39rs3r6 s 4 4 2 3 4 2 39 tw9 tw4(2h2 k)(7h 1 k2 )29 Using Scienti c NotationRecall at the beginning of the section that we found the number 1.3 1013 when describing bits of informationin digital images. Other extreme numbers include the width of a human hair, which is about 0.00005 m, andthe radius of an electron, which is about 0.00000000000047 m. How can we e ectively work read, compare,and calculate with numbers such as these?A shorthand method of writing very small and very large numbers is calledscienti c notation, in whichwe express numbers in terms of exponents of 10. To write a number in scienti c notation, move the decimalpoint to the right of the rst digit in the number. Write the digits as a decimal number between 1 and 10.Count the number of places n that you moved the decimal point. Multiply the decimal number by 10 raisedto a power of n. If you moved the decimal left as in a very large number, n is positive. If you moved thedecimal right as in a small large number, n is negative.For example, consider the number 2,780,418. Move the decimal left until it is to the right of the rstnonzero digit, which is 2.We obtain 2.780418 by moving the decimal point 6 places to the left. Therefore, the exponent of 10 is 6, and it ispositive because we moved the decimal point to the left. This is what we should expect for a large number.2.780418 106(29)Working with small numbers is similar. Take, for example, the radius of an electron, 0.00000000000047 m.Perform the same series of steps as above, except move the decimal point to the right.http://cnx.org/content/m51241/1.8/
OpenStax-CNX module: m5124114Be careful not to include the leading 0 in your count. We move the decimal point 13 places to the right, so theexponent of 10 is 13. The exponent is negative because we moved the decimal point to the right. This iswhat we should expect for a small number.4.7 10 13A number is written inA General Note:10n ,where 1 a 10 and n isscienti c notation if it is written in the form aan integer.Example 10Converting Standard Notation to Scienti c NotationWrite each number in scienti c notation.a. Distance to Andromeda Galaxy from Earth: 24,000,000,000,000,000,000,000 mb. Diameter of Andromeda Galaxy: 1,300,000,000,000,000,000,000 mc. Number of stars in Andromeda Galaxy: 1,000,000,000,000d. Diameter of electron: 0.00000000000094 me. Probability of being struck by lightning in any single year: 0.00000143Solutiona.24, 000, 000, 000, 000, 000, 000, 000m24, 000, 000, 000, 000, 000, 000, 000m 222.4 1022b.placesm1, 300, 000, 000, 000, 000, 000, 000m1, 300, 000, 000, 000, 000, 000, 000m 211.3 1021placesm1, 000, 000, 000, 000c.1, 000, 000, 000, 000 12places1 1012d.0.00000000000094m0.00000000000094m9.4 13 places 10 13 mhttp://cnx.org/content/m51241/1.8/(30)
OpenStax-CNX module: m51241150.00000143e.0.00000143 6places1.43 10 6AnalysisObserve that, if the given number is greater than 1, as in examples a c, the exponent of 10 ispositive; and if the number is less than 1, as in examples d e, the exponent is negative.Try It:Exercise 20(Solution on p. 25.)Write each number in scienti c notation.a.U.S. national debt per taxpayer (April 2014): 152,000b.World population (April 2014): 7,158,000,000c.World gross national income (April 2014): 85,500,000,000,000d.Time for light to travel 1 m: 0.00000000334 se.Probability of winning lottery (match 6 of 49 possible numbers): 0.00000007159.1 Converting from Scienti c to Standard NotationTo convert a number in scienti c notation to standard notation,simply reverse the process. Move thedecimal n places to the right if n is positive or n places to the left if n is negative and add zeros as needed.Remember, if n is positive, the value of the number is greater than 1, and if n is negative, the value of thenumber is less than one.Example 11Converting Scienti c Notation to Standard NotationConvert each number in scienti c notation to standard notation.a.b.c.d.3.547 1014 2 1067.91 10 7 8.05 10 12Solution3.547 1014a.3.54700000000000 14places354, 700, 000, 000, 000 2 106b. 2.000000 6places 2, 000, 000http://cnx.org/content/m51241/1.8/
OpenStax-CNX module: m51241167.91 10 7c.0000007.91 7places0.000000791 8.05 10 12 000000000008.05d. 12places 0.00000000000805Try It:Exercise 22(Solution on p. 25.)Convert each number in scienti c notation to standard notation. 10511b. 8.16 10 13c. 3.9 10 6d.8 10a.7.039.2 Using Scienti c Notation in ApplicationsScienti c notation, used with the rules of exponents, makes calculating with large or small numbers mucheasier than doing so using standard notation. For example, suppose we are asked to calculate the number ofatoms in 1 L of water. Each water molecule contains 3 atoms (2 hydrogen and 1 oxygen). The average drop of 1021 molecules of water and 1 L of water1.22 104 average drops. holds about25214approximately 3 · 1.32 10· 1.22 10 4.83 10 atoms in 1 L of water.water contains around 1.32Therefore, there areWe simply multiply the decimal terms and add the exponents. Imagine having to perform the calculationwithout using scienti c notation!When performing calculations with scienti c notation, be sure to write the answer in proper scienti cnotation. For example, consider the productscienti c notation because 35 is greater than 7 104 · 5 106 35 1010 . The answer is not in proper10. Consider 35 as 3.5 10. That adds a ten to the exponentof the answer.(35) 1010 (3.5 10) 1010 3.5 10 1010 3.5 1011Example 12Using Scienti c NotationPerform the operations and write the answer in scienti c notation.a.b.c.d.e. 78.14 106.5 1010 4 105 10 9 1.52 1352.7 10 6.04 10 1.2 108 9.6 105 3.33 104 1.05 107 5.62 105http://cnx.org/content/m51241/1.8/(31)
OpenStax-CNX module: m5124117Solution8.14 10 7 6.5 1010 (8.14 6.5) 10 7 1010Commutative and associative properties of multiplicationa. (52.91) 10 4 105 1.52 109 b. 4 1.525.291 10 5 ( 2.63) 102.7 1056.04 1013 Product rule of exponents4Scienti c notationCommutative and associative10109 4 3 2.63 10properties of multiplication Quotient rule of exponents 4Scienti c notation(2.7 6.04) 105 1013Commutative and associative properties of multiplicationc. (16.308) 10 1.2 108 9.6 105 d. 3.33 10 7 1.05 10e. 1.6308 10 8 Product rule of exponents19Scienti c notationCommutative and associative10105properties of multiplication3(0.125) 10 Quotient rule of exponents21.25 10 5.62 105 41.29.618Scienti c notation [3.33 ( 1.05) 5.62] 104 107 105 ( 19.65) 1016 1.965 1017Try It:Exercise 24(Solution on p. 25.)Perform the operations and write the answer in scienti c notation.a.b.c.d.e. 7.5 108 1.13 10 2 1.24 1011 1.55 10183.72 109 8 103 9.933 1023 2.31 1017 6.04 109 7.3 102 2.81 102Example 13Applying Scienti c Notation to Solve ProblemsIn April 2014, the population of the United States was about 308,000,000 people. The nationaldebt was about 17,547,000,000,000. Write each number in scienti c notation, rounding gures totwo decimal places, and nd the amount of the debt per U.S. citizen. Write the answer in bothscienti c and standard notations.http://cnx.org/content/m51241/1.8/
OpenStax-CNX module: m51241Solution18 3.08 108 .13was 17, 547, 000, 000, 000 1.75 10 .The population was 308, 000, 000The national debtTo nd the amount of debt per citizen, divide the national debt by the number of citizens. 1.75 1013 3.08 108The debt per citizen at the time was about 5.7 1013 · 108 1.753.08 0.57 105 5.7 104 104 ,or(32) 57,000.Try It:Exercise 26(Solution on p. 25.)An average human body contains around 30,000,000,000,000 red blood cells. Each cellmeasures approximately 0.000008 m long. Write each number in scienti c notation andnd the total length if the cells were laid end-to-end. Write the answer in both scienti cand standard notations.Media:Access these online resources for additional instruction and practice with exponents andscienti c notation. Exponential Notation1Properties of ExponentsZero Exponent234Simplify Exponent ExpressionsQuotient Rule for ExponentsScienti c Notation567Converting to Decimal Notation1 http://openstaxcollege.org/l/exponnot2 http://openstaxcollege.org/l/exponprops3 http://openstaxcollege.org/l/zeroexponent4 http://openstaxcollege.org/l/exponexpres5 http://openstaxcollege.org/l/quotofexpon6 http://openstaxcollege.org/l/scienti cnota7 .org/content/m51241/1.8/
OpenStax-CNX module: m512411910 Key EquationsRules of ExponentsFor nonzero real numbers a and b and integers m and nProduct ruleQuotient ruleam · an am namm nan am nm·nPower rule(a ) aZero exponent rulea0 1Negative rulea n Power of a product rule(a · b) an · bn na n abnbPower of a quotient rule1annTable 111 Key Concepts Products of exponential expressions with the same base can be simpli ed by adding exponents. SeeExample 1. Quotients of exponential expressions with the same base can be simpli ed by subtracting exponents.See Example 2. Powers of exponential expressions with the same base can be simpli ed by multiplying exponents. SeeExample 3. An expression with exponent zero is de ned as 1. See Example 4.An expression with a negative exponent is de ned as a reciprocal. See Example 5 and Example 6.The power of a product of factors is the same as the product of the powers of the same factors. SeeExample 7. The power of a quotient of factors is the same as the quotient of the powers of the same factors. SeeExample 8. The rules for exponential expressions can be combined to simplify more complicated expressions. SeeExample 9. Scienti c notation uses powers of 10 to simplify very large or very small numbers. See Example 10 andExample 11. Scienti c notation may be used to simplify calculations with very large or very small numbers. SeeExample 12 and Example 13.12 Section Exercises12.1 VerbalExercise 273(Solution on p. 25.)2Is 2 the same as 3? Explain.Exercise 28When can you add two exponents?Exercise 29What is the purpose of scienti c notation?Exercise 30Explain what a negative exponent does.http://cnx.org/content/m51241/1.8/(Solution on p. 25.)
OpenStax-CNX module: m512412012.2 NumericFor the following exercises, simplify the given expression. Write answers with positive exponents.Exercise 31(Solution on p. 25.)92Exercise 3215 2Exercise 33(Solution on p. 25.)32 33Exercise 3444 4Exercise35 2 2(Solution on p. 25.)2Exercise 360(5 8)Exercise 37(Solution on p. 25.)113 114Exercise 3865 6 7Exercise39 80(Solution on p. 25.)2Exercise 405 2 52For the following exercises, write each expression with a single base. Do not simplify further. Write answerswith positive exponents.Exercise 41(Solution on p. 25.)42 43 4 4Exercise 4261269Exercise 43 123 12(Solution on p. 25.)10Exercise 44 2106 1010Exercise 45(Solution on p. 25.)7 6 7 3Exercise 46533 34For the following exercises, express the decimal in scienti c notation.Exercise 47(Solution on p. 25.)0.0000314Exercise 48148,000,000For the following exercises, convert each number in scienti c notation to standard notation.Exercise 491.6 1010http://cnx.org/content/m51241/1.8/(Solution on p. 26.)
OpenStax-CNX module: m5124121Exercise 509.8 10 912.3 AlgebraicFor the following exercises, simplify the given expression. Write answers with positive exponents.Exercise 51(Solution on p. 26.)a3 a2aExercise 52mn2m 2Exercise53 (Solution on p. 26.)2b3 c4Exercise 5 54 x 3y2Exercise 55(Solution on p. 26.)ab2 d 3Exercise 156w 0 x5Exercise 57(Solution on p. 26.)m4n0Exercise 258y 4 y 2Exercise 59p 4 q 2p2 q 3(Solution on p. 26.)Exercise 602(l w)Exercise61 y73(Solution on p. 26.) x14Exercise62 2a23Exercise 63(Solution on p. 26.)52 m 50 mExercise64 (16x)y 12Exercise 65(Solution on p. 26.)23(3a) 2Exercise 2 661m3 a2ma6Exercise67 3bc3Exercise 68 2x2 y 13 y 0http://cnx.org/content/m51241/1.8/(Solution on p. 26.)
OpenStax-CNX module: m5124122Exercise69 9z3 2(Solution on p. 26.)y12.4 Real-World ApplicationsExercise 70To reach escape velocity, a rocket must travel at the rate of 2.2 106 ft/min.Rewrite the rate instandard notation.Exercise 71(Solution on p. 26.)A dime is the thinnest coin in U.S. currency. A dime's thickness measures 1.35 10 3 m.Rewritethe number in standard notation.Exercise 72The average distance between Earth and the Sun is 92,960,000 mi.Rewrite the distance usingscienti c notation.Exercise 73(Solution on p. 26.)A terabyte is made of approximately 1,099,500,000,000 bytes. Rewrite in scienti c notation.Exercise 74The Gross Domestic Product (GDP) for the United States in the rst quarter of 2014 was 1.71496 1013 . Rewritethe GDP in standard notation.Exercise 75One picometer is approximately 3.397 10 11 in.(Solution on p. 26.)Rewrite this length using standard notation.Exercise 76The value of the services sector of the U.S. economy in the rst quarter of 2012 was 10,633.6billion. Rewrite this amount in scienti c notation.12.5 TechnologyFor the following exercises, use a graphing calculator to simplify. Round the answers to the nearest hundredth.Exercise77 123 m334 3(Solution on p. 26.)2Exercise 78173 152 x312.6 ExtensionsFor the following exercises, simplify the given expression. Write answers with positive exponents.Exercise79 23a3 24a22Exercise 8062 24 2 (Solution on p. 26.)2 5xyExercise 81m2 n3a2 c 3·a 7 n 2m2 c4http://cnx.org/content/m51241/1.8/(Solution on p. 26.)
OpenStax-CNX module: m5124123Exercise82 10 x6 y 3x3 y 3·y 7x 3Exercise83 2 3(ab c)(Solution on p. 26.)2b 3Exercise 84Avogadro's constant is used to calculate the number of particles in a mole.A mole is a basicunit in chemistry to measure the amount of a substance. The constant is 6.0221413 1023 . WriteAvogadro's constant in standard notation.Exercise 85(Solution on p. 26.)Planck's constant is an important unit of measure in quantum physics. It describes the relationshipbetween energy and frequency.
Exponents and Scientific Notation * OpenStax OpenStax Algebra and Trigonometry This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 Abstract In this section students will: Use the product rule of exponents. Use the quotient rule of exponents. Use the power rule of exponents.
It's Practice with Scientific Notation! Review of Scientific Notation Scientific notation provides a place to hold the zeroes that come after a whole number or before a fraction. The number 100,000,000 for example, takes up a lot of room and takes time to write out, while 10 8 is much more efficient.File Size: 290KBPage Count: 8People also search forscientific notation worksheet answersscientific notation worksheet keyscientific notation worksheet pdf answersscientific notation worksheet with answersscientific notation worksheetscientific notation worksheet with answer key
Recall the definition for scientific notation 1. Change these LARGE scientific notation numbers to standard notation and vice versa. Make up a number for the blank cells. Scientific Notation Standard Notation Scientific Notation Standard Notation a. 6.345 10 e. 5,320 b. 8.04 10 % f. 420,000 c. 4.26 10 & g. 9,040,000,000 d. h. 2. Now try .
scientific notation. Operations in Scientific Notation 1. Perform the calculations on the “number” parts in the front of the scientific notation numbers. 2. Use rules of exponents on the 10n parts of the numbers in scientific notation. 3. Make sure your answer is in scientific notation, if
4. Write 4321000 in scientific notation. (Remember that scientific notation requires multiplying a number between 1 and 10 by a power of 10.) 5. Write 0.065 in scientific notation. Hint: this requires negative exponents! 6. Write these numbers without exponents: a. 7.8 · 106 b. 7.8 · 10-6 7. Write these numbers in scientific notation: a. 9012 .
Scientific Notation (SN)- A shorthanded way of writing really large or really small numbers. In SN a number is written as the product of two factors. !Ex: 280,000,000 can be written in scientific notation as 2.8!!!10. First Factor Regular Notation ! Scientific Notation Regular Notation How to Change Scientific Notation 420,000.
known as the scientific notation. If A is a number between 1 and 10 or 1 and n is an integer, then A x10n is a number written in scientific notation (1 A 10) Examples 1).write 80 000 in scientific notation 80 000 8 x 10 000 8 x 104 2).Write 354 in scientific notation 354 3.54 x 100 3.54 102 3).Write 63.33 in scientific notation 63. .
Express 0.5 in scientific notation Express 0.72 in scientific notation Write 0.05 in scientific notation 1. Denote the decimal numbers in the first column in scientific notation and complete the table given below. 2. Write in scientific notation (i) 0.9 (ii) 0.08 (iii) 0.05 (iv) 0.032 (v) 0.00021 Decimal number in general form
Animal Food Nutrition Science Public Health Sports & Exercise Healthcare Medical 2.3 Separate, speciality specific listings providing examples of the detailed areas of knowledge and application for each of the five new core competencies required by Registered Nutritionist within these specialist areas have been created and are listed later in this document under the relevant headings. 2.4 All .
