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 = y n , then x is the n th root of y.The principal n th 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} }
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)125 -
25 2-2 -
93 / 95 -
03 -
( 2 / y)5 -
(- 3)4 -
(2 / 5) - 1 -
| - 2 |4 -
(-3)0 -
(- 1)4 -
(- 1)15 -
(32)3 -
(- 4 x)3 -
\sqrt[4]{16^3} -
27^{5/3} -
\sqrt[3]{32} \cdot \sqrt[3]{2} -
\dfrac{\sqrt{160}}{\sqrt{40}} -
(\sqrt[6]{ 3})^6 -
\sqrt[4]{ (- 7)^4 } -
\sqrt[5]{(- 9)^5}
Answers to Above Questions
-
(-1)125 = - 1 -
25 2-2 = 23 -
93 / 95 = 9-2 = 1 / 92 -
03 = 0 -
( 2 / y)5 = 32 / y5 -
(- 3)4 = 34 -
(2 / 5) - 1 = 5 / 2 -
| - 2 |4 = 24 -
(-3)0 = 1 -
(- 1)4 = 1 -
(- 1)15 = - 1 -
(32)3 = 3 6 -
(- 4 x)3 = (- 4)3 x3 = - 43 x3 -
\sqrt[4]{16^3} = (\sqrt[4]{16})^3 = 2^3 -
27^{5/3} = (\sqrt[3]{27})^5 = 3^5 -
\sqrt[3]{32} \cdot \sqrt[3]{2} = \sqrt[3]{64} = 4 -
\dfrac{\sqrt{160}}{\sqrt{40}} = \sqrt{\dfrac{160}{40}} = \sqrt{4} = 2 -
(\sqrt[6]{ 3})^6 = 3 -
\sqrt[4]{ (- 7)^4 } = | - 7 | = 7 -
\sqrt[5]{(- 9)^5} = - 9
More References and links
Simplify Radical ExpressionsWhat are Exponents in Maths