Exercise Simplify – 22. – 4

Exercise Simplify (– 2)2. 4

Exercise Simplify – 23. – 8

Exercise Simplify (– 2)3. – 8

Exercise Explain the difference between – x2 and (– x)2. The negative sign is not taken to the power in the first expression, but it is in the second.

A positive real number has two square roots, one positive and one negative.

16 is a perfect square.

A number is a perfect cube if it is the result of cubing an integer.

8 is a perfect cube.

radical sign index √ 8 = 2 3 radicand

Differences between Square Roots and Cube Roots of Real Numbers Any real number has exactly one cube root. But any positive real number has two square roots, one positive and one negative.

Differences between Square Roots and Cube Roots of Real Numbers 2. The cube root of a negative number is a negative number. But there is no real square root of a negative number.

(– 2)3 √ – 8 3

√ – 64 0 real roots √ 64 2 real roots √ – 64 1 real root 3 3 √ 64 1 real root

Example 1 3 Find √ 125. Since 53 = 125, the radical √ 125 = 5. 3

Since (– 6)3 = – 216, the radical √ – 216 = – 6. Example 1 3 Find √ – 216. Since (– 6)3 = – 216, the radical √ – 216 = – 6. 3

Example True or false: – 53 = (– 5)3 True

Example True or false: – 122 = (– 12)2 False

Example True or false: – 1331 = (– 13)31 True

Example 2 Find the consecutive integers between which √ 70 lies. Use the < symbol to express your answers. 3 43 = 64 and 53 = 125. Since 70 lies between 64 and 125, 4 < √ 70 < 5. 3

Example 2 Find the consecutive integers between which √ – 5 lies. Use the < symbol to express your answers. 3 (– 1)3 = – 1 and (– 2)3 = – 8. Since – 5 lies between – 8 and – 1, – 2 < √ – 5 < – 1. 3

Example Between which two consecutive integers does √ 12 lie? 2 and 3

For all a ≥ 0 and b ≥ 0 with n ≥ 0, √ a • √ b = √ ab. Product Law for Roots For all a ≥ 0 and b ≥ 0 with n ≥ 0, √ a • √ b = √ ab. n

Example 4 Simplify √ 2,700. √ 2,700 = √ 2 • 2 • 3 • 3 • 3 • 5 • 5 = √ 33 • √ 2 • 2 • 5 • 5 3 = 3 √ 2 • 2 • 5 • 5 = 3 3 √ 100 = 3

Example 5 3 Simplify 7 √ 1,080. 7 √ 1,080 3 = 7 √ 2 • 2 • 2 • 3 • 3 • 3 • 5 3 7 √ 23 • √ 33 • √ 5 = 3 = 7 • 2 • 3 √ 5 3 42 √ 5 = 3

Example 3 3 Simplify √ 432 and √ 50,625. 6 √ 2; 15 √ 15 3

Example 3 3 Simplify √ 18 • √ 24. 6 √ 2 3

Example 3 3 Simplify √ 135 – √ 40. √ 5 3

Exercise 3 3 Simplify √ 12 • √ 18. 6

Exercise 3 3 Simplify √ – 140 √ 150. – 10 √ 21 3

Exercise 3 3 Simplify √ 24 + √ 81. 5 √ 3 3

Exercise 3 3 3 Simplify 3 √ 16 + 5 √ 40 – √ 54. 3 √ 2 + 10 √ 5 3

Exercise 3 Which is larger, √ 11 or √ 11? √ 11