Exponents
Author Ter-Petrosyan Hakob
Product Rule
When you multiply powers with the same base, keep the base and add the exponents:
\[a^m \times a^n = a^{m + n}\]Example: Multiply \(x^3\) by \(x^4\):
\[x^3 \times x^4 = (x \times x \times x) \times (x \times x \times x \times x) = x^7\]Use this rule to make expressions with the same base simpler by adding the exponents instead of writing out each factor.
The Quotient Rule of Exponents
For any real number a and natural numbers m and n such that m > n, the quotient rule of exponents states that
\[\frac{a^m}{a^n} = a^{m-n}\]The Power Rule of Exponents
For any real number a and positive integers m and n, the power rule of exponents states that
\[(a^m)^n=a^{m \cdot n}\]The Zero Exponent Rule of Exponents
For any nonzero real number a, the zero exponent rule of exponents states that
\[a^0 = 1\]The Negative Rule of Exponents
For any nonzero real number a and natural number n the negative rule of exponents states that
\[a^{-n}= \frac{1}{a^n}\]The Power of a Product Rule of Exponents
For any real numbers a and b and any integer n, the power of a product rule of exponents states that
\[(ab)^n = a^n \cdot b^n\]The Power of a Quotient Rule of Exponents
For any real numbers a and b and any integer n, the power of a quotient rule of exponents states that
\[\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}\]