6.3 Approximating Square Roots - Big Ideas Learning
English6.3SpanishApproximating Square RootsHow can you find decimal approximations ofSSTATESTANDARDSMA.8.A.6.2square roots that are irrational?You already know that a rational number is a number that can be written as theratio of two integers. Numbers that cannot be written as the ratio of two integersare called irrational.Real NumbersRationalIntegerNatural21312 1 2 3Irrationalπ2 322.25 30.31 2πACTIVITY: Approximating Square RootsWork with a partner.Archimedes was a Greek mathematician, physicist,engineer, inventor, and astronomer.a. Archimedes tried to find a rational numberwhose square is 3. Here are two that he tried.2651531351780— and ——Are either of these numbers equal to 3 ?How can you tell?Archimedes(c. 287 B.C.–c. 212 B.C.)b. Use a calculator with —a square rootkey to approximate 3 .Write the number on a piece ofpaper. Then enter it into thecalculator and square it. Thensubtract 3. Do you get 0? Explain.c. Calculators did not exist inthe time of Archimedes. Howdo you think he —might haveapproximated 3 ?244Chapter 6Square Roots and the Pythagorean TheoremSquareRoot Key
EnglishSpanish2ACTIVITY: Approximating Square Roots GeometricallyWork with a partner.a. Use grid paper and the given scale to draw a horizontal line segment1 unit in length. Label this segment AC.b. Draw a vertical line segment 2 units in length. Label this segment DC.c. Set the point of a compass on A. Set the compass to 2 units. Swing thecompass to intersect segment DC. Label this intersection as B.d. Use the PythagoreanTheorem to show that the length of—segment BC is 3 units.—e. Use the grid paper to approximate 3 .DB22Scale:110C1of a unitA3. Repeat Activity 2 for a triangle in which segment CA is 2 units andsegment BA is 3 —units. Use the Pythagorean Theorem to show—thatsegment BC is 5 units. Use the grid paper to approximate 5 .4. IN YOUR OWN WORDS How can you find decimal approximations ofsquare roots that are irrational?Use what you learned about approximating square roots tocomplete Exercises 5 – 8 on page 249.Section 6.3Approximating Square Roots245
English6.3SpanishLessonLesson TutorialsA rational number is a number that can be written as the ratio of twointegers. An irrational number cannot be written as the ratio of two integers.Key Vocabularyirrational number,p. 246real numbers, p. 246 The square root of any whole number that is not a perfect squareis irrational. The decimal form of an irrational number neither terminatesnor repeats.Real NumbersRational numbers and irrational numbers together form the set ofreal numbers.Real NumbersRationalIntegerRememberDecimals that terminateor repeat are rational.Natural21312 1 2 3Irrationalπ2 322.25 30.3EXAMPLE 2πClassifying Real Numbers1Tell whether the number is rational or irrational. Explain.Numbera.— 12Rational or IrrationalReasoningIrrationalRational12 is not a perfect square.— 0.364 is a repeating decimal.—b. 0.364c. 1—37Rational 1— can be written as —.d.0.85Rational0.85 can be written as —. 107371720Tell whether the number is rational or irrational. Explain.Exercises 9–14246Chapter 61. 0.121221222. . .2.— 196Square Roots and the Pythagorean Theorem3.— 2
EnglishSpanishEXAMPLE2Approximating Square Roots—Estimate 52 to the nearest integer.Use a number line and the square roots of the perfect squares nearest tothe radicand. The nearest perfect square less than 52 is 49. The nearestperfect square greater than 52 is 64.Graph52 .49 764 8—Because 52 is closer to 49 than to 64, 52 is closer to 7 than to 8.—So, 52 7.Estimate to the nearest integer.—EXAMPLE—4. 33Exercises 18–233—5. 85—6. 1907. 7Comparing Real Numbers—34a. Which is greater, 5 or 2 —?Graph the numbers on a number line.2 3 2.75544 29 3—34342 — is to the right of 5 . So, 2 — is greater.——b. Which is greater, 0.6 or 0.36 ?Graph the numbers on a number line.0.36 0.60.60.60.7———0.6 is to the right of 0.36 . So, 0.6 is greater.Which number is greater? Explain.Exercises 25–3015—8. 4 —, 239.—— 10 , 5Section 6.310.— 2 , 2Approximating Square Roots247
EnglishSpanishEXAMPLE4Approximating an Expression —The radius of a circle with area A is approximatelyA3— . The area of acircular mouse pad is 51 square inches. Estimate its radius.circul —A— 3—513—Substitute 51 for A.— 17Divide.The nenearest perfect square less than 17 is 16. The nearest perfect squaregreater than 17 is 25.1725 516 4—Because 17 is closer to 16 than to 25, 17 is closer to 4 than to 5.The radius is about 4 inches.11. WHAT IF? The area of a circular mouse pad is 64 square inches.Estimate its radius.EXAMPLE5Real-Life ApplicationThe distance (in nautical miles) you can—see with a periscope is 1.17 h , whereh is the height of the periscope abovethe water. Can a periscope that is 6 feetabove the water see twice as far as aperiscope that is 3 feet above the water?Explain.hUse a calculator to find the distances.b6 ffeet abovewater3 feet above water——1.17 h 1.17 3 2.03Substitute for h.Use a calculator.——1.17 h 1.17 6 2.872.872.03You can see — 1.41 times farther with the periscope that is 6 feetabove the water than with the periscope that is 3 feet above the water.No, the periscope that is 6 feet above the water cannot see twice as far.12. You use a periscope that is 10 feet above the water. Can yousee farther than 4 nautical miles? Explain.248Chapter 6MSFL8PE 0603.indd 248Square Roots and the Pythagorean Theorem10/20/09 4:27:21 PM
EnglishSpanishExercises6.3Help with Homework1. VOCABULARY What is the difference between a rational number and anirrational number?—2. WRITING Describe a method of approximating 32 .3. VOCABULARY What are real numbers? Give three examples.4. WHICH ONE DOESN’T BELONG? Which number does not belong with theother three? Explain your reasoning.1112 —— 3.3— 825.0756) 39 (- 3) 3 (- 9) 4 (- 1)9 (-Tell whether the rational number is a reasonable approximation of the square root.559250—5. —, 5—3021250678250—16772507. —, 286. —, 11—8. —, 45Tell whether the number is rational or irrational. Explain.1—9. 3.66666612. 1.125π6—11. 710. ——8914. 1513. 3 — 15. ERROR ANALYSIS Describe and correct theerror in classifying the number.— 144 is irrational.16. SCRAPBOOKING You cut a picture into a right triangle for your scrapbook.The lengths of the legs of the triangle are 4 inches and 6 inches. Is the lengthof the hypotenuse a rational number? Explain.Real NumbersRationalIntegerNaturalIrrational17. VENN DIAGRAM Place each number inthe correct area of the Venn Diagram.a. Your ageb. The square root of any prime numberc. The ratio of the circumference of a circleto its diameterSection 6.3Approximating Square Roots249
EnglishSpanishEstimate to the nearest integer.———19. 6852 18. 24 20. 61 ——21. 10522.—274335223. ——24. CHECKERS A checkerboard is 8 squares long and 8 squares wide. The areaof each square is 14 square centimeters. Estimate the perimeter of thecheckerboard.Which number is greater? Explain.——3 25. 20 , 10228. —,3 26. 15 , 3.5——168129. 0.25 , 0.25——3427. 133 , 10 ———30. 182 , 19231. FOUR SQUARE The area of afour square court is 66 squarefeet. Estimate the length s of oneof the sides of the court.ss32. RADIO SIGNAL The maximum distance(in nautical miles) that a radio transmittersignal can be sent—is represented by theexpression 1.23 h , where h is the height(in feet) above the transmitter.Estimate the maximum distance x(in nautical miles) between the plane thatis receiving the signal and the transmitter.Round your answer to the nearest tenth.x22,000 ftNot drawn to scale33. OPEN-ENDED Find two numbers a and b that satisfy the diagram.9250Chapter 6ab10Square Roots and the Pythagorean Theorem
EnglishSpanishEstimate to the nearest tenth.——34. 0.39r â 16.764 m—35. 1.1936. 1.5237. ROLLER COASTER The velocity v (in meters per second)of a rollercoaster is represented by the equation—v 3 6r , where r is the radius of the loop. Estimatethe velocity of a car going around the loop. Roundyour answer to the nearest tenth. —38. Is1— a rational number? Is4 —316— a rational number? Explain.39. WATER BALLOON The time t (in seconds) it takes a waterballoon to fall d meters is represented by the equation —d4.9t — . Estimate the time it takes the balloon to fallto the ground from a window that is 14 meters abovethe ground. Round your answer to the nearest tenth.40.Determine if the statement is sometimes, always,ways,or never true. Explain your reasoning and give an example of each.a. A rational number multiplied by a rational number is rational.ationalb. A rational number multiplied by an irrational number is rational.c. An irrational number multiplied by an irrational number is rational.Simplify the expression. SKILLS REVIEW HANDBOOK42. 3π 8(t π) 4t41. 2x 3y 5x43. 17k 9 23k44. MULTIPLE CHOICE What is the ratio (red to blue) of the corresponding sidelengths of the similar triangles? SKILLS REVIEW HANDBOOK20822.59A 1:3 104B 5:2 C 3:4 Section 6.3D 2:5 Approximating Square Roots251
Section 6.3 Approximating Square Roots 247 EXAMPLE 2 Approximating Square Roots Estimate — 52 to the nearest integer. Use a number line and the square roots of the perfect squares nearest to the radicand. The nearest perfect square less than 52 is 49. The nearest perfect square greater than 52 is 64. Graph 52 . 49 7 64 8
Roots of complex numbers Every number has two square roots. The square roots of 16 are: The square roots of 24 are: The square roots of -81 are: The square roots of -75 are: Likewise, every number has three cube roots, four fourth roots, etc. (over the complex number system.) So if we want to find the four fo
When approximating square roots of numbers, you need to determine: · Between which two perfect squares it lies (and therefore which 2 square roots). · Which perfect square it is closer to (and therefore which square root). Example: Lies between 100 & 121, closer to 100. So is between 10 & 11, closer to 10. Square Perfect Root Square
Grade 8 - Unit 1 Square roots & Pythagorean Theorem Name: _ Estimating Square Roots It is important to be able to estimate square roots of a number. To estimate the square root of a number ; 1. Write out the first few perfect squar
Section 4.3 Solving Quadratic Equations Using Square Roots 211 Solving a Quadratic Equation Using Square Roots Solve (x 1)2 25 using square roots.SOLUTION (x 1)2 25 Write the equation.x 1 5 Take the square root of each side. x 1 5 Add 1 to each side. So, the solutions are x 1 5 6 and x 1 5 4. Check Use a graphing calculator to check
Find the square root(s). 4. — 1 5. — 4 — 25 6. — 12.25 EXAMPLE 2 Finding Square Roots Exercises 7–18 — 2.25 represents both the positive and the negative square roots. — 25 represents the positive square root. — represents the 9 — 16 negative square root.
Find the square roots. 1. 9 2. 36 3. 121 Simplify. 4. 1 5. 36 6. 7. 0.0064 81 100 Answers on page A-43 Objectives Find principal square roots and their opposites, approximate square roots, find outputs of square-root functions, graph square-root functions, and find the domains of square-root functions. Simplify radical expressions with perfect .
Category A: Estimating Square Roots and Cube Roots . Between what two consecutive integers . do the following real numbers lie between? 5 38 53 99 326 3214 227 77 171 194 380 147 3999 3119 380. Category B: Square Roots and Cube
down your commitment to practice jazz piano, tell it to others, and schedule in specific practice times. MONTH ONE: Jazz Piano 101 A. Chord types (Play each in all keys) 2 B. Quick Fix Voicing C. ETUDE: (Quick fix voicings with inversions for better voice leading) ALL MUSICAL EXAMPLES TAKEN FROM “JAZZ PIANO HANDBOOK” (ALFRED PUBLISHING) AND USED WITH PERMISSION MONTH TWO: Position .
