# 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.
## 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.
## QuestionsUse 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} -
2^{5}2^{-2} -
9^{3}/ 9^{5} -
0^{3} -
( 2 / y)^{5} -
(- 3)^{4} -
(2 / 5)^{ - 1} -
| - 2 |^{4} -
(-3)^{0} -
(- 1)^{4} -
(- 1)^{15} -
(3^{2})^{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 -
2^{5}2^{-2}= 2^{3} -
9^{3}/ 9^{5}= 9^{-2}= 1 / 9^{2} -
0^{3}= 0 -
( 2 / y)^{5}= 32 / y^{5} -
(- 3)^{4}= 3^{4} -
(2 / 5)^{ - 1}= 5 / 2 -
| - 2 |^{4}= 2^{4} -
(-3)^{0}= 1 -
(- 1)^{4}= 1 -
(- 1)^{15}= - 1 -
(3^{2})^{3}= 3^{6} -
(- 4 x)^{3}= (- 4)^{3}x^{3}= - 4^{3}x^{3} -
\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 linksSimplify Radical ExpressionsWhat are Exponents in Maths |