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📘intermediate algebra review

Exponent Laws

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

Exponent laws are the set of rules that govern the manipulation and simplification of expressions involving exponents. These laws provide a systematic way to perform operations with exponents, such as multiplication, division, and raising an exponent to another exponent, which are essential in the context of dividing radical expressions.

5 Must Know Facts For Your Next Test

  1. The product rule states that when multiplying terms with the same base, the exponents are added: $a^m \cdot a^n = a^{m+n}$.
  2. The quotient rule states that when dividing terms with the same base, the exponents are subtracted: $\frac{a^m}{a^n} = a^{m-n}$.
  3. The power rule states that when raising a term with an exponent to another exponent, the exponents are multiplied: $(a^m)^n = a^{m \cdot n}$.
  4. The zero exponent rule states that any base raised to the power of zero is equal to 1: $a^0 = 1$.
  5. The negative exponent rule states that $a^{-n} = \frac{1}{a^n}$, which allows for the simplification of expressions with negative exponents.

Review Questions

  • Explain how the product rule can be used to simplify the expression $3^4 \cdot 3^2$.
    • According to the product rule, when multiplying terms with the same base, the exponents are added. In the expression $3^4 \cdot 3^2$, the base is 3 in both terms. Applying the product rule, we can simplify the expression as $3^{4+2} = 3^6$.
  • Describe how the quotient rule can be used to simplify the expression $\frac{x^8}{x^3}$.
    • The quotient rule states that when dividing terms with the same base, the exponents are subtracted. In the expression $\frac{x^8}{x^3}$, the base is x in both the numerator and denominator. Applying the quotient rule, we can simplify the expression as $x^{8-3} = x^5$.
  • Analyze the steps required to simplify the expression $(2^3)^4$ using the power rule.
    • The power rule states that when raising a term with an exponent to another exponent, the exponents are multiplied. In the expression $(2^3)^4$, the base is $2^3$, and the exponent is 4. Applying the power rule, we can simplify the expression as $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$.