# 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 $$0^ 0$$   is undefined $$0^m = 0$$ , $$m \gt 0$$ $$0^{10} = 0$$ $$x^0 = 1 \; , \; x \ne 0$$ $$21^0 = 1$$ $$1^m = 1$$ $$1^{12} = 1$$ $$(-1)^m = 1$$ if $$m$$ is even integer $$(-1)^6 = 1$$ $$(-1)^m = - 1$$ if $$m$$ is odd integer $$(-1)^9 = - 1$$ $$x^m x^n = x^{m+n}$$ $$2^3 2^4 = 2^{3 + 4} = 2^7$$ $$\dfrac{x^m}{x^n} = x^{m-n}$$ $$\dfrac{7^5}{7^2} = 7^{5-2} = 7^3$$ $$(x^m)^n = x^{m \times n}$$ $$(4^5)^2= 4^{5 \times 2} = 4^{10}$$ $$(x \; y)^m = x^m \; y^m$$ $$(3 \; b)^2 = 3^2 \; b^2 = 9 b^2$$ $$\left ( \dfrac{x}{y} \right )^m = \dfrac{x^m}{y^m} \; , y \ne 0$$ $$\left ( \dfrac{4}{b} \right )^2 = \dfrac{4^2}{b^2} = \dfrac{16}{b^2}$$ $$(-x)^m = (-1)^m \; x^m$$ $$(-3)^4 = (-1)^4 (3^4) = 1 \times 81 = 81$$ $$\left (\dfrac{x}{y} \right)^{-m} = \left (\dfrac{y}{x} \right)^{m} \; , \; x \ne 0 \; , y \; \ne 0$$ $$\left (\dfrac{4}{3} \right)^{-2} = \left (\dfrac{3}{4} \right)^{2}$$ $$\dfrac{1}{y^{-m}} = y^m \; , \; y \ne 0$$ $$\dfrac{1}{8^{-2}} = 8^2$$ $$| x^m| = | x |^m$$ $$| (- 5)^4 | = |-5| ^4 = 5^4 = 625$$

## What are Radicals?

If $$x = y^n$$ , then $$x$$ is the $$n^{th}$$ root of $$y$$.
The
principal $$n^{nt}$$ root of a number $$x$$ 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 $$\sqrt[n]{x^m} = (\sqrt[n]{x})^m$$ $$\sqrt[3]{27^2} = (\sqrt[3]{27})^2 = 3^2$$ $$x^{m/n} = (\sqrt[n]{x})^m = \sqrt[n]{x^m}$$ $$4^{3/2} = (\sqrt{4})^3 = 2^3$$ $$\sqrt[n]{x} \cdot \sqrt[n]{y} = \sqrt[n]{x y}$$ $$\sqrt[5]{16} \cdot \sqrt[5]{2} = \sqrt[5]{16 \times 2} = \sqrt[5]{32} = 2$$ $$\dfrac{\sqrt[n]{x}}{\sqrt[n]{y}} = \sqrt[n]{\dfrac{x}{y}}$$ $$\dfrac{\sqrt[3]{-40}}{\sqrt[3]{5}} = \sqrt[3]{\dfrac{-40}{5}} = \sqrt[3]{-8} = - 2$$ $$(\sqrt[m]{x})^m = x$$ $$(\sqrt[3]{ - 2})^3 = - 2$$ $$\sqrt[m]{x^m} = | x | \;\; \text{if m is even}$$ $$\sqrt[4]{(- 2)^4} = | - 2| = 2$$ $$\sqrt[m]{x^m} = x \;\; \text{if m is odd}$$ $$\sqrt[5]{ (- 2)^5} = - 2$$

### 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. $$2^5 \; 2^{-2}$$

3. $$9^3 / 9^5$$

4. $$0^3$$

5. $$( 2 / y)^5$$

6. $$(- 3)^4$$

7. $$(2 / 5)^{-1}$$

8. $$| - 2 |^4$$

9. $$(-3)^0$$

10. $$(- 1)^4$$

11. $$(- 1)^{15}$$

12. $$(3^2)^3$$

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. $$2^5 \; 2^{-2} = 2^3 = 8$$

3. $$9^3 / 9^5 = 1 / 9^2 = 1/81$$

4. $$0^3 = 0$$

5. $$( 2 / y)^5 = 2^5 / y^5 = 32 / y^5$$

6. $$(- 3)^4 = 3^4 = 81$$

7. $$(2 / 5)^{-1} = 5/2$$

8. $$| - 2 |^4 = 2^4 = 16$$

9. $$(-3)^0 = 1$$

10. $$(- 1)^4 = 1$$

11. $$(- 1)^{15} = -1$$

12. $$(3^2)^3 = 3^6 = 729$$

13. $$(- 4 x)^3 = (- 4)^3 x^3 = - 4^3 \; x^3 = - 64 x^3$$

14. $$\sqrt[4]{16^3} = (\sqrt[4]{16})^3 = 2^3 = 8$$

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. $$(\sqrt[6]{ 3})^6 = 3^{(6/6)} = 3^1 = 3$$

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

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