The exponent rules are very helpful formulas to use when you are simplifying exponential expressions. They are also called the "exponent laws" or the "exponent properties".
These rules are used to simplify advanced exponential expressions.
I have a different page that explains how to simplify basic exponents, if you're learning exponents for the first time.
You can use them to simplify expressions that have just numbers, just variables, or some combination of both. You can also use them in reverse.
If you click on the "Learn More" links below, you'll see step-by-step examples and explanations that show WHY each rule works the way it does.
I highly recommend reading those pages because it is much easier to memorize a formula when you know WHY it works.
\(x^{0}=1\)
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\(x^{-a}=\frac{1}{x^{a}}\)
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\(x^{\frac{a}{b}}=\sqrt[b]{x^a}\)
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\(x^{a}x^{b}=x^{a+b}\)
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\(\frac{x^{a}}{x^{b}}=x^{a-b}\)
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\((x^{a})^{b}=x^{ab}\)
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\((xy)^a=x^a y^a\)
\((\frac{x}{y})^a=\frac{x^a}{y^a}\)
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Each of the rules above can be used in reverse. For example...
Write \(\sqrt[4]{9^2}\) in exponential form.
This expression is a power within a root, so I could use the fraction exponents rule in reverse to write it as...
\(9^{\frac{2}{4}}\)
\(\frac{2}{4}\) can be reduced to \(\frac{1}{2}\) so the simplified answer in exponential form is...
\(9^{\frac{1}{2}}\)
The reverse distributive property of exponents is used so often that some people call it the "product rule for powers with the same exponent".
The reverse power rule of exponents is also used quite often, especially for changing the units in exponential functions.
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