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Exponents

An exponent of a number indicates how many times to multiply the number by itself. Exponents are also called powers or indices.

For example: Let’s take a look at the number 95. Here 9 is called the base and 5 is the exponent ( or power or index). You would usually say “9 to the power of 5” or “9 to the fifth exponent”. It tells us that 9 has to be multiplied five times by 9, that is: 95=9·9·9·9·9

Properties of exponents

PropertyExplanation

a^n=a*a*...*a
an is a multiplied by itself n times

a^{-n}=frac{1}{a^n}

a-1 is the same as 1 divided by an

a^0=1

a0 is by definition always 1

a^n*a^p=a^{n+p}

Two numbers with the same bases but different exponents
are multiplied together by adding the exponents

a^n:a^p=a^{n-p}

Two numbers with the same bases but different exponents
are divided by each other by subtracting the exponents
from each other.

a^x*b^x=(a*b)^x

Two numbers with the same exponents but different
bases are multiplied together by multiplying the bases
and then raise the result to the exponent

(a^n)^p=a^{n*p}

If a number with an exponent is the base to an exponent,
the two exponents are multiplied together

a^{frac{1}{2}}=sqrt{a}
A base, which is raised to the exponent one half,
is equal to the square root of the base.

a^{frac{1}{3}}=sqrt[3]{a}

A base, which is raised to the exponent one third,
is equal to the cubic root of the base.

sqrt[n]{a^p}=a^{frac{p}{n}}

The n-th root of the base a raised to the exponent p
is equal to the base a raised to the exponent p/n