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Math help from the Learning Centre

This guide provides useful resources for a wide variety of math topics. It is targeted at students enrolled in a math course or any other Centennial course that requires math knowledge and skills.

What is an exponent?

The exponent of a number indicates the number of times the base number is multiplied by itself. For example,

\[2^4=2\times 2\times 2\times 2 \qquad \left(\frac{1}{2}\right)^2 = \frac{1}{2} \times \frac{1}{2}\]


Laws of exponents: These laws help you simplify exponents.

\[b^n \cdot b^m =  b^{n+m}\]

When multiplying by the same base, the exponents can be added together. For example,

\[3^4 \cdot 3^2 = \left(3\times 3\times 3\times 3\right) \cdot \left(3\times 3\right) = \left(3\times 3\times 3\times 3\times 3\times 3\right) =3^{4+2} = 3^6\]

 

\[b^n \div b^m =  b^{n-m}\]

When dividing by the same base, the exponents can be subtracted, top minus bottom. For example,

\[\frac{5^4}{5^2} = \frac{5\times 5\times 5\times 5}{5\times 5} = 5 ^{4-2} = 5^2\]

 

\[\left(b^n\right)^m = b^{n\times m}\]

When multiple exponents are on a base, the exponents can be multiplied together. For example,

\[\left(7^2\right)^3 = \left(7\times 7\right)^3 = \left(7\times 7\right)\left(7\times 7\right)\left(7\times 7\right) = 7^{2\times 3} = 7^6\]

Special Exponents

Negative Exponents:

A negative exponent signals a reciprocal of the base and exponents. Once the reciprocal is performed, the negative exponent becomes positive. 

\[2^{-3}=\frac{1}{2^3}=\frac{1}{8}\]

This can also be interpreted as moving a value from the top of the fraction to the bottom and vice versa.

\[\frac{1}{4^{-3}}=\frac{4^3}{1}=64\qquad \frac{3^{-5}}{2^{-2}} = \frac{2^{2}}{3^{5}} = \frac{4}{243} \]


Power of 1:

Any base to the exponent of 1 equals itself.

\[13^1=13 \qquad \pi^1=\pi \qquad (2.2)^1=2.2\]


Power of 0:

Any non-zero number with the exponent 0 equals to 1.

\[5^0=1 \qquad x^0=1 \qquad \left(\frac{17}{19}\right)^0=1\]

Do you know the reason why?

Creative Commons License
Designed by Matthew Cheung. This work is licensed under a Creative Commons Attribution 4.0 International License.
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