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