Types of Exponents

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)(1,2,3)

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

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

Rational Numbers: An exponent that is a fraction (12,34,56)(\frac{1}{2}, \frac{3}{4}, \frac{5}{6} )

Counting Number Exponents

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

If you have an base aa raised to the exponent bb such that aba^{b}, then your are multiplying aa by 11, bb times.

a1=1×aa^{1}=1 \times a

a2=1×a×aa^{2}=1 \times a \times a

a5=1×a×a×a×a×aa^{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 aa raised to the exponent 00, then you multiply aa by 11 zero times, leaving you with 11. Any number to the exponent zero is just 11.


Negative Integer Exponents

When You have a base aa raised to a negative integer exponent bb such that aba^{-b} you divide 11 by aa, bb times.

a1=1÷aa^{-1}= 1 \div a

a1=1aa^{-1}= \frac{1}{a}

a2=1÷a÷aa^{-2}= 1 \div a \div a

a2=1a2a^{-2}= \frac{1}{a^{2}}

a5=1÷a÷a÷a÷a÷aa^{-5}= 1 \div a \div a \div a \div a \div a

a5=1a5a^{-5}= \frac{1}{a^{5}}

Rational Exponents

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

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


6423 64^{\frac{2}{3}}

=6423= \sqrt[3]{64^{2}} OR (643)2 (\sqrt[3]{64})^{2}

=40963= \sqrt[3]{4096} OR 42 4^{2}


Note by Brody Acquilano
6 years, 3 months ago

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not bad............

kanya shah - 6 years, 2 months ago

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