Exponent Rules
Product rule
\[a^m \cdot a^n = a^{m+n}\]
Quotient Rule
\[a^m \div a^n = a^{m - n}\]
Power of a Power Rule
\[(a^m)^n = a^{m \cdot n}\]
Power of a Product Rule
\[(ab)^m = b^m \cdot b^m\]
Power of a Quotient Rule
\[(\frac{a}{b})^m = \frac{a^m}{b^m}\]
Zero Exponent Rule
\[a^0 = 1\]
Negative Exponent Rule
\[a^{-m} = \frac{1}{a^m}\]
Fractional Exponent Rule
\[a^{\frac{m}{n}} = \sqrt[n]{a^m}\]
Odd vs Even exponents and negative coefficients
Even Exponents
\[(-x)^4\]
In this case this has a coefficients of negative one. so it is equavilant to
\[x^4\]
Odd Exponents
\[3(-x)^3\]
This is equivalant to
\[-3x^3\]
Multiplying Negative Exponents
When the bases are the same
\[a^{-n} \cdot a^{-m} = a^{-(n+m)} = 1 \div a^{n+m}\]
When the bases are different and the negative powers are the same
\[a^{-n} \cdot b^{-n} = (a \cdot b)^{-n} = 1 \div (a \cdot b)^n\]
When the bases and the negative powers are different
\[a^{-n} \cdot b^{-m} = (a^{-n}) \cdot (b^{-m})\]