Power of a Power Property
The Power of a Power Property in Algebra 1 (California Reveal Math, Grade 9) states that when a power is raised to another power, multiply the exponents: (aᵐ)ⁿ = aᵐⁿ. For example, (x³)⁴ = x¹² and (2²)⁵ = 2¹⁰ = 1024. This works because (aᵐ)ⁿ means multiplying aᵐ by itself n times, adding the exponents each time. This property is one of the core exponent rules used constantly in simplifying polynomial expressions, scientific notation, and exponential functions throughout Algebra 1 and beyond.
Key Concepts
When a power is raised to another power, multiply the exponents:.
$$(a^m)^n = a^{m \cdot n}$$.
Common Questions
What is the Power of a Power Property?
When a power is raised to another power, multiply the exponents: (aᵐ)ⁿ = aᵐⁿ. For example, (x³)⁴ = x¹².
Why do you multiply exponents instead of adding them?
You multiply because (aᵐ)ⁿ means using aᵐ as a factor n times: aᵐ × aᵐ × ... × aᵐ (n times) = a^(m+m+...+m) = a^(m×n).
Is (aᵐ)ⁿ the same as aᵐⁿ?
Yes, (aᵐ)ⁿ = aᵐⁿ = a^(m·n). The order of multiplication does not matter: (aᵐ)ⁿ = aᵐⁿ = (aⁿ)ᵐ.
How does the Power of a Power rule differ from the Product of Powers rule?
Power of a Power: (aᵐ)ⁿ = aᵐⁿ (exponents multiplied — one base in parentheses). Product of Powers: aᵐ × aⁿ = aᵐ⁺ⁿ (exponents added — two identical bases being multiplied).
Where is the Power of a Power Property covered in California Reveal Math Algebra 1?
This property is taught in California Reveal Math, Algebra 1, as part of Grade 9 exponent rules and polynomial operations.
Can you apply this rule to negative exponents?
Yes. (a⁻²)³ = a^(-2×3) = a⁻⁶. The rule works for any integer exponents, positive, negative, or zero.
What common mistake do students make with this rule?
Students often add exponents instead of multiplying, confusing Power of a Power with Product of Powers.