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Chapter 6 25 Glencoe Algebra 2

Simplify Radicals

Simplify √ �� 49z8 .

√ �� 49z8 = √ ��� (7z4)2 = 7z4

z4 must be positive, so there is no need to take the absolute value.

Simplify - 3 √ ���� (2a - 1)6

- 3 √ ���� (2a - 1)6 = - 3 √ ����� [(2a - 1)2]3 = -(2a - 1)2

ExercisesSimplify.

1. √ �� 81 2. 3 √ ��� -343 3. √ ��� 144p6

4. ± √ �� 4a10 5. 5 √ ��� 243 p 10 6. - 3 √ �� m6n9

7. 3 √ �� -b12 8. √ ��� 16a10b8 9. √ ��� 121x6

10. √ �� (4k)4 11. ± √ ��� 169r4 12. - 3 √ ��� -27 p 6

13. - √ ��� 625y2z4 14. √ ��� 36q34 15. √ ���� 100x2y4z6

16. 3 √ ��� -0.027 17. - √ ��� -0.36 18. √ ��� 0.64p10

19. 4 √ �� (2x)8 20. √ ��� (11y2)4 21. 3 √ ��� (5a2b)6

22. √ ���� (3x - 1)2 23. 3 √ ���� (m - 5)6 24. √ ������ 36x2 - 12x + 1

Square Root For any real numbers a and b, if a2 = b, then a is a square root of b.

nth Root For any real numbers a and b, and any positive integer n, if an = b, then a is an n th

root of b.

Real nth Roots of b,

n √ � b , - n √ � b

1. If n is even and b > 0, then b has one positive real root and one real negative root.

2. If n is odd and b > 0, then b has one positive real root.

3. If n is even and b < 0, then b has no real roots.

4. If n is odd and b < 0, then b has one negative real root.

6-4 Study Guide and Interventionnth Roots

Example 1 Example 2

9 -7 12|p3|

±2a5 3p2 -m 2n3

-b4 4|a5|b4 11|x 3|

16k 2 ±13r 2 3p 2

-25|y |z 2 6|q17| 10|x |y 2|z 3|

-0.3 not a real number 0.8|p5|

4x2 121y 4 25a4b2

|3x - 1| (m - 5)2 |6x - 1|

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Chapter 6 26 Glencoe Algebra 2

Approximate Radicals with a Calculator

Radicals such as √ � 2 and √ � 3 are examples of irrational numbers. Decimal approximations for irrational numbers are often used in applications. These approximations can be easily found with a calculator.

Use a calculator to approximate 5 √ �� 18.2 to three decimal places.

5 √ �� 18.2 ≈ 1.787

ExercisesUse a calculator to approximate each value to three decimal places.

1. √ �� 62 2. √ �� 1050 3. 3 √ ��� 0.054

4. - 4 √ �� 5.45 5. √ �� 5280 6. √ ��� 18,600

7. √ ��� 0.095 8. 3 √ �� -15 9. 5

√ �� 100

10. 6 √ �� 856 11. √ �� 3200 12. √ �� 0.05

13. √ ��� 12,500 14. √ �� 0.60 15. - 4 √ �� 500

16. 3 √ �� 0.15 17. 6

√ �� 4200 18. √ �� 75

19. LAW ENFORCEMENT The formula r = 2 √ �� 5L is used by police to estimate the speed r in miles per hour of a car if the length L of the car’s skid mark is measures in feet. Estimate to the nearest tenth of a mile per hour the speed of a car that leaves a skid mark 300 feet long.

20. SPACE TRAVEL The distance to the horizon d miles from a satellite orbiting h miles above Earth can be approximated by d = √ ����� 8000h + h2 . What is the distance to the

horizon if a satellite is orbiting 150 miles above Earth?

Irrational Number a number that cannot be expressed as a terminating or a repeating decimal

Study Guide and Intervention (continued)

nth Roots

6-4

Example

7.874 32.404 0.378

-1.528 72.664 136.382

0.308 -2.466 2.512

3.081 56.569 0.224

111.803 0.775 -4.729

0.531 4.017 8.660

77.5 mi/h

about 1100 mi

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