The Power of a Power Rule
Unpack (2^3)^2 to discover why we multiply exponents, not add them.
Equation
y = 2^(1*x)
Graph
Table
| x | y |
|---|---|
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
| 4 | 16 |
What this lesson covers
What you do
You shape the function y = b^(m*x) and watch the graph answer.
Challenges to clear
- First look inside the bracket: with b = 2 and m = 3, the x = 1 row shows 2^3 = 8.
- Set b=2 and m=3. Find the value of y when x=2. This demonstrates that (2^3)^2 = 2^(3*2) = 64.
- Start from a base of 3 — slide b there.
- Now slide m until x = 2 reads 81. That is (3²)² = 3⁴ — powers of powers MULTIPLY, they do not add.
Check yourself
Why does the rule (a^m)^n = a^(m*n) work?
Using the power of a power rule, what is the simplified form of (3^2)^3?
- Because raising a power to another power creates n groups of m factors, resulting in m*n total factors. — correct
- Because the outer exponent adds to the inner exponent.
- Because multiplying bases requires adding exponents.
- Because the rule is just a convention we memorize without logical basis.
- 3^5
- 3^6 — correct
- 9^3
- 3^8