# Rules for Radicals and Exponents

Important rules to simplify radical expressions and expressions with exponents are presented along with examples. Questions with answers are at the bottom of the page.

## Rules for Exponents.

 Rules Examples 00 is undefined 0m = 0 , m > 0 010 = 0 x0 = 1 , x ≠ 0 210 = 1 1m = 1 112 = 1 (-1)m = 1 if m is even integer (-1)6 = 1 (-1)m = - 1 if m is odd integer (-1)9 = - 1 xm xn = xm + n 23 24 = 23 + 4 = 27 xm / xn = xm - n 75 / 72 = 75 - 2 = 73 (xm)n = xm × n (45)2 = 45 × 2 = 410 (x y)m = xm y m (3 b)2 = 32 b2 (x / y)m = xm / y m , y ≠ 0 (4 / b)2 = 4 2 / b2 (- x)m = (-1)m x m (- 3)4 = (-1)4 34 = 34 (x / y) - m = ym / x m , x ≠ 0 , y ≠ 0 (4 / 3) - 2 = (3 / 4) 2 1 / y - m = ym , y ≠ 0 (1 / 8) - 2 = 8 2 | x m| = | x | m = x m if m is even integer | - 5 |4 = (-5) 4

## What are Radicals?

If x = yn, then x is the nth root of y.
The
principal nth root x of a number has the same sign as x.
Examples
1) The square (second) root of 4 is 2 (Note: - 2 is also a root but it is not the principal because it has opposite site to 4)
2) The cube (third) root of 8 is 2
4) The cube (third) root of - 8 is - 2
Special symbols called
radicals are used to indicate the principal root of a number.
\huge \color{red}{ y = \sqrt[n]{x} }
n is the index, x is the radicand. For the square root (n = 2), we dot write the index.

## Rules for Radicals.

 Rules Examples \sqrt[n]{x^m} = (\sqrt[n]{x})^m \sqrt[3]{27^2} = (\sqrt[3]{27})^2 = 3^2 x^{m/n} = (\sqrt[n]{x})^m = \sqrt[n]{x^m} 4^{3/2} = (\sqrt{4})^3 = 2^3 \sqrt[n]{x} \cdot \sqrt[n]{y} = \sqrt[n]{x y} \sqrt[5]{16} \cdot \sqrt[5]{2} = \sqrt[5]{32} = 2 \dfrac{\sqrt[n]{x}}{\sqrt[n]{y}} = \sqrt[n]{\dfrac{x}{y}} \dfrac{\sqrt[3]{-40}}{\sqrt[3]{5}} = \sqrt[3]{\dfrac{-40}{5}} = \sqrt[3]{-8} = - 2 (\sqrt[m]{x})^m = x (\sqrt[3]{ - 2})^3 = - 2 \sqrt[m]{x^m} = | x | \;\; \text{if m is even} \sqrt[4]{(- 2)^4} = | - 2| = 2 \sqrt[m]{x^m} = x \;\; \text{if m is odd} \sqrt[5]{ (- 2)^5} = - 2

### Questions

Use the rules listed above to simplify the following expressions and rewrite them with positive exponents. Note that sometimes you need to use more than one rule to simplify a given expression.

1. (-1)125

2. 25 2-2

3. 93 / 95

4. 03

5. ( 2 / y)5

6. (- 3)4

7. (2 / 5) - 1

8. | - 2 |4

9. (-3)0

10. (- 1)4

11. (- 1)15

12. (32)3

13. (- 4 x)3

14. \sqrt[4]{16^3}

15. 27^{5/3}

16. \sqrt[3]{32} \cdot \sqrt[3]{2}

17. \dfrac{\sqrt{160}}{\sqrt{40}}

18. (\sqrt[6]{ 3})^6

19. \sqrt[4]{ (- 7)^4 }

20. \sqrt[5]{(- 9)^5}

### Answers to Above Questions

1. (-1)125 = - 1

2. 25 2-2 = 23

3. 93 / 95 = 9-2 = 1 / 92

4. 03 = 0

5. ( 2 / y)5 = 32 / y5

6. (- 3)4 = 34

7. (2 / 5) - 1 = 5 / 2

8. | - 2 |4 = 24

9. (-3)0 = 1

10. (- 1)4 = 1

11. (- 1)15 = - 1

12. (32)3 = 3 6

13. (- 4 x)3 = (- 4)3 x3 = - 43 x3

14. \sqrt[4]{16^3} = (\sqrt[4]{16})^3 = 2^3

15. 27^{5/3} = (\sqrt[3]{27})^5 = 3^5

16. \sqrt[3]{32} \cdot \sqrt[3]{2} = \sqrt[3]{64} = 4

17. \dfrac{\sqrt{160}}{\sqrt{40}} = \sqrt{\dfrac{160}{40}} = \sqrt{4} = 2

18. (\sqrt[6]{ 3})^6 = 3

19. \sqrt[4]{ (- 7)^4 } = | - 7 | = 7

20. \sqrt[5]{(- 9)^5} = - 9