Properties of Exponents and Simplifying Expressions

Properties of Exponents

Negative Exponents and Power of a Quotient

Understanding how to simplify expressions with exponents is a fundamental skill in College Algebra. This topic covers the rules for negative exponents and how to apply them to quotients and powers.

• Negative Exponent Rule: For any nonzero number x and integer n, $x^{-n} = \frac{1}{x^n}$.

• Power of a Quotient Rule: For any nonzero numbers a and b, and integer n, $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$.

• Power of a Power Rule: $\left(x^m\right)^n = x^{mn}$.

Example: Simplifying the Expression

Given the expression:

$\left(\frac{a^2}{8}\right)^{-2}$

• Apply the negative exponent rule:

• $\left(\frac{a^2}{8}\right)^{-2} = \left(\frac{8}{a^2}\right)^2$

• Apply the power of a quotient rule:

• $\left(\frac{8}{a^2}\right)^2 = \frac{8^2}{(a^2)^2}$

• Simplify further:

• $8^2 = 64$

• $(a^2)^2 = a^{2 \times 2} = a^4$

• Final answer:

$\left(\frac{a^2}{8}\right)^{-2} = \frac{64}{a^4}$

Key Points

• Negative exponents indicate reciprocals.

• When raising a quotient to a power, raise both numerator and denominator to that power.

• When raising a power to another power, multiply the exponents.

Additional info:

• These rules are foundational for simplifying algebraic expressions and solving equations in College Algebra.