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Rules of Exponents

Here are some basic rules for operations with exponents:

\[ \begin{align} x^m \times x^n &= x^{m+n} \\
\frac{x^m}{x^n} &= x^{m-n} \\
x^{-n} &= \frac{1}{x^n} \\
(x^m)^n &= x^{mn} \\
(xy)^{n} &= x^n y^n \\
\left(\frac{x}{y}\right)^m &= \frac{x^m}{y^m} \end{align} \]

Radicals are just numbers with fractional indices, e.g. \( \sqrt[3]{7^2} = 7^{2/3} \). Any operation with exponents can be applied to radicals, and they are related through this rule:

\[ x^{m/n} = \sqrt[n]{x^m} \]

Note by Arron Kau
3 years ago

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Thanks for reminding me. Shraman Das · 2 years, 12 months ago

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I have an issue with the answer to the solution for the problem in this site. ((-5)^10)^(3/10)=125. Why should it not be +-125? I need an explanation. Arnab Laha · 2 years, 11 months ago

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@Arnab Laha Arnab, you have to perform operations in parentheses first. Arron Kau Staff · 2 years, 11 months ago

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@Arron Kau Is that so easy? Not for me at-least. What is (9765625)^(1/10)? Will it be +5 or -5 or +-5. Lets think about it carefully? :) Arnab Laha · 2 years, 10 months ago

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@Arnab Laha In the absence of reasons otherwise or clear specification, convention is to assume that we're looking for the principle root (i.e., the positive root, in this case). Arron Kau Staff · 2 years, 10 months ago

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thanks for the note.. Shashwata Samanta · 2 years, 11 months ago

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