3. The power of a power
Exercises
Definitions
The marking [[$ (3^2)^4 $]] stands for the power of a power. The exponent is the number 4 and the base number is the number inside the brackets, i.e. [[$3^2$]]. Let us consider the power as a base number like a single number. The power expression can be written as follows:
[[$ (3^2)^4 = \underbrace{3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2}_{\text{This is a product of powers with the same base that is already known.}} = 3 \overbrace{^{2+2+2+2}}^{=2 \cdot 4} =3^8 $]]
[[$ \left(a^m\right)^n = a^{m \cdot n}$]]
[[$ (3^2)^4 = \underbrace{3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2}_{\text{This is a product of powers with the same base that is already known.}} = 3 \overbrace{^{2+2+2+2}}^{=2 \cdot 4} =3^8 $]]
The power of a power
A power is raised to a power by multiplying the exponents. The base stays the same.[[$ \left(a^m\right)^n = a^{m \cdot n}$]]
Examples
Example 1
Simplify the powers.
| a) | [[$ (2^3)^2 = 2^{3 \cdot 2} = 2^6 = 64 $]] | |
| b) | [[$ 16(a^2)^2 = 16 \cdot a^{2 \cdot 2} = 16a^4 $]] | Only [[$a^2$]] is the base of the power. |
| c) | [[$ 2^{3^2} = 2^9 =512 $]] |
This is not a power of power! |
Example 2
To what power does number [[$3$]] need to be raised in order for the value to be equal to number [[$ 9^5 $]]? In other words, what number can replace [[$ x $]] in the following equation: [[$ 3^x = 9^5 $]]? Solution:
The base number of power [[$ 9^5 $]] is [[$ 9 $]], which is obtained as the power of number three as follows: [[$ 3^2 = 9 $]]. By replacing the number nine with this power and simplifying the calculation, we arrive at the result [[$ 9^5 = (3^2)^5 = 3^{2\cdot 5} = 3^{10} $]].
Answer: The number [[$3$]] must be raised to power [[$10$]].