Power of a Power Property
The Power of a Power Property states that to raise an expression already raised to a power to another power, keep the base and multiply the exponents: (a^m)^n = a^(mn). Taught in Yoshiwara Elementary Algebra Chapter 9: More About Exponents and Roots, this rule helps Grade 6 students simplify expressions like (x³)⁴ = x¹² quickly. It is frequently used alongside the product and quotient rules when simplifying complex algebraic expressions.
Key Concepts
Property To raise a power to a power, keep the same base and multiply the exponents. In symbols, $$(a^m)^n = a^{mn}$$.
Examples To simplify $(x^3)^5$, you multiply the exponents: $x^{3 \cdot 5} = x^{15}$. To simplify $(4^2)^3$, you keep the base and multiply the powers: $4^{2 \cdot 3} = 4^6$. Be careful to distinguish from products: $(a^5)(a^2) = a^{5+2} = a^7$, but $(a^5)^2 = a^{5 \cdot 2} = a^{10}$.
Explanation Think of this as repeated multiplication. $(x^4)^3$ is just $x^4$ multiplied by itself three times. Adding the exponents $4+4+4$ is the same as multiplying $4 \cdot 3$. So, you multiply the exponents.
Common Questions
What is the power of a power property?
When raising a power to another power, keep the base and multiply the exponents: (a^m)^n = a^(mn). For example, (x³)⁴ = x^12.
How is the power of a power different from the product rule?
The product rule adds exponents when multiplying powers: a^m × a^n = a^(m+n). The power of a power rule multiplies exponents when raising a power to a power: (a^m)^n = a^(mn).
Can the power of a power rule apply to negative exponents?
Yes. For example, (a^(-2))^3 = a^(-6), which simplifies to 1/a^6.
Where is the power of a power property in Yoshiwara Elementary Algebra?
It is in Chapter 9: More About Exponents and Roots of Yoshiwara Elementary Algebra.
How do you simplify (2^3)^2?
Multiply the exponents: (2^3)^2 = 2^(3×2) = 2^6 = 64.