The Power of a Power Rule for Exponents
The power of a power rule for exponents is a Grade 8 math skill covered in Chapter 7: Exponents and Scientific Notation. The rule states that to raise a power to another power, multiply the exponents: (a^m)^n = a^(m*n). This rule simplifies expressions where an exponent itself is raised to another exponent.
Key Concepts
Property To raise a power to another power, you multiply the exponents. The general form is $(a^m)^n = a^{m \cdot n}$.
Examples $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$ $(10^5)^2 = 10^{5 \cdot 2} = 10^{10}$ $(x^6)^3 = x^{6 \cdot 3} = x^{18}$.
Explanation The power of a power rule simplifies expressions where an exponential term is raised to another exponent. For example, $(2^3)^4$ means you are multiplying $2^3$ by itself four times: $2^3 \cdot 2^3 \cdot 2^3 \cdot 2^3$. Using the product rule, you would add the exponents: $2^{3+3+3+3} = 2^{12}$. This rule provides a shortcut by simply multiplying the two exponents, $3 \cdot 4$, to get the same result.
Common Questions
What is the power of a power rule for exponents?
(a^m)^n = a^(m x n). When raising a power to another power, multiply the exponents.
What is (x^3)^4?
(x^3)^4 = x^(3 x 4) = x^12.
What is (2^3)^2?
(2^3)^2 = 2^(3 x 2) = 2^6 = 64.
Where is the power of a power rule taught in Grade 8?
Chapter 7: Exponents and Scientific Notation in 8th grade math.
Why do you multiply exponents when raising a power to a power?
(a^m)^n means multiplying a^m by itself n times, which means using the base a a total of m x n times as a factor.