Power of a power
Grade 8 math lesson on the power of a power rule for exponents: (a^m)^n = a^(m times n). Students learn to simplify nested exponential expressions by multiplying the exponents when a power is raised to another power.
Key Concepts
Property To find a power of a power, multiply the exponents. $$(x^a)^b = x^{ab}$$.
Examples To simplify $(x^5)^3$, we multiply the exponents: $x^{5 \cdot 3} = x^{15}$. This works for numbers too: $(2^4)^2 = 2^{4 \cdot 2} = 2^8$. Even with nested powers: $((y^2)^3)^4 = (y^6)^4 = y^{6 \cdot 4} = y^{24}$.
Explanation This is like having groups of groups! The outer exponent tells you how many times to repeat the inner power's entire group of factors. Instead of writing it all out, which would take forever, you can use the ultimate shortcut: just multiply the two exponents together to find the grand total number of factors in your super powered expression.
Common Questions
What is the power of a power rule for exponents?
When a power is raised to another power, multiply the exponents: (a^m)^n = a^(mn). For example, (x^3)^4 = x^12, because 3 x 4 = 12.
How is the power of a power rule different from multiplying powers?
When multiplying powers with the same base (x^3 times x^4), you add exponents to get x^7. When raising a power to a power (x^3)^4, you multiply exponents to get x^12. The two rules are often confused.
How do you simplify (2^3)^2?
(2^3)^2 = 2^(3x2) = 2^6 = 64. Using the power of a power rule, multiply the exponents 3 and 2 to get 6, then evaluate 2^6.
When does the power of a power rule apply?
The power of a power rule applies when you have an exponential expression inside parentheses that is raised to an outer exponent, like (x^5)^3. Multiply the inner and outer exponents.