Exponent Properties

When working with exponents that have the same base, you can use several key properties to simplify expressions. For multiplication, you add the exponents when multiplying terms with the same base $e.g., a^5 · a^3 = a^8$. This extends to algebraic expressions like 2g^5h^7$$11g^2h = 22g^7h^8.

Division works similarly, but you subtract the exponents (top minus bottom). For example, m^10 ÷ m^3 = m^7, and fractions like 18w^5z^7 ÷ $-9wz^6$ simplify to -2w^4z. When dealing with powers of expressions, use the power rule by multiplying the exponents: $x^3y^5$^7 = x^21y^35.

Remember that anything raised to the power of zero equals 1 $e.g., 3^0 = 1$, and negative exponents move to the opposite side of the fraction line while becoming positive $x^-4 = 1/x^4$. These rules can be combined to tackle complex expressions like $-3a^4b^7$^2$6a^5b$ ÷ $-2a^7b^2$ = 27a^6b^13.

Pro Tip: When simplifying expressions with multiple exponent rules, always apply the power rule first, then handle multiplication and division. This order prevents common mistakes!

With rational exponents, the numerator represents the power and the denominator represents the root. For example, 81^$3/4$ means (∜81)^3, and expressions like (√25)^-3 = 25^$-3/2$ = 1/125.