**Types of Exponents and Types of Numbers**

Before we get into doing operations with exponents lets make sure we understand the different types of exponents. There are many types of exponents as there are many types of numbers. I will list the basic ones that will provide you with the understanding to get through most problems. Keep in mind that negative and rational exponents can be combined.

*Counting Numbers:* An exponent that is number you can count like if you were counting sheep. \((1,2,3)\)

*Zero exponents:* An exponent that is zero. \((0)\)

*Negative Integers:* An exponent that is a negative number but is not a fraction. \((-1,-2,-3)\)

*Rational Numbers:* An exponent that is a fraction \((\frac{1}{2}, \frac{3}{4}, \frac{5}{6} )\)

**Counting Number Exponents**

Lets start off by explaining the counting number exponents \((1,2,3)\).

If you have an base \(a\) raised to the exponent \(b\) such that \(a^{b}\), then your are multiplying \(a\) by \(1\), \(b\) times.

\(a^{1}=1 \times a\)

\(a^{2}=1 \times a \times a\)

\(a^{5}= 1\times a \times a \times a \times a \times a\)

**Zero Exponents**

What if you have an exponent that is zero. Well, if you just read the section on counting numbers and asked why there was a one out front, it will probably make sense now. If you have a base \(a\) raised to the exponent \(0\), then you multiply \(a\) by \(1\) zero times, leaving you with \(1\). Any number to the exponent zero is just \(1\).

\(a^{0}=1\)

**Negative Integer Exponents**

When You have a base \(a\) raised to a negative integer exponent \(b\) such that \(a^{-b}\) you divide \(1\) by \(a\), \(b\) times.

\(a^{-1}= 1 \div a\)

\(a^{-1}= \frac{1}{a}\)

\(a^{-2}= 1 \div a \div a\)

\(a^{-2}= \frac{1}{a^{2}}\)

\(a^{-5}= 1 \div a \div a \div a \div a \div a\)

\(a^{-5}= \frac{1}{a^{5}}\)

**Rational Exponents**

Rational, or fractional exponents turn into radicals or roots. If there is an expression \(a^{\frac{b}{c}}\) then you find the \(c\) root of \(a\) and then raise it to the exponent \(b\). \(c\) is reffered to as the index which tells you which root you are taking.

The general rule is that \(a^{\frac{b}{c}}=\sqrt[c]{a^{b}}\)

example

\( 64^{\frac{2}{3}} \)

\(= \sqrt[3]{64^{2}}\) OR \( (\sqrt[3]{64})^{2} \)

\(= \sqrt[3]{4096} \) OR \( 4^{2} \)

\(=16 \)

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