Power of a Power Property
Apply the Power of a Power Property in Grade 9 algebra: when raising a power to another power, multiply the exponents—(xᵐ)ⁿ=xᵐⁿ—to simplify nested exponential expressions efficiently.
Key Concepts
Property If $m$ and $n$ are real numbers and $x \neq 0$, then $(x^m)^n = x^{mn}$.
Examples To simplify $(3^2)^4$, you multiply the exponents: $3^{2 \cdot 4} = 3^8$. For a variable, the rule is the same: $(x^5)^3 = x^{5 \cdot 3} = x^{15}$. This also works with coefficients: $(( 2r)^2)^3 = ( 2r)^{2 \cdot 3} = ( 2r)^6 = 64r^6$.
Explanation Think of this as stacking powers! When a power is raised to another power, you're just doing repeated multiplication. Instead of writing it all out, you can take a shortcut and simply multiply the two exponents together. This makes simplifying expressions with 'nested' exponents super quick and easy to handle.
Common Questions
What is the Power of a Power Property?
The Power of a Power Property states (xᵐ)ⁿ = xᵐⁿ. When a power is raised to another power, multiply the exponents. For example, (x³)⁴ = x¹² because 3 × 4 = 12.
How do you apply Power of a Power with coefficients, like (2x³)⁴?
Apply the exponent to both the coefficient and the variable separately. Raise the coefficient to the power: 2⁴ = 16. Multiply variable exponents: x³ raised to 4 = x¹². Result: 16x¹².
How does Power of a Power differ from the Product Rule?
Product Rule (xᵐ · xⁿ = xᵐ⁺ⁿ) applies when multiplying two powers with the same base — add exponents. Power of a Power ((xᵐ)ⁿ = xᵐⁿ) applies when a power is raised to another power — multiply exponents.