No. 8 Calculations with powers

Find the value of an expression with powers. Bring the powers to one base, and everything cancels.

How to solve it

Expressions with powers

An expression like $\frac{2^5 \cdot 4^3}{8^3}$ looks heavy, but once every number is written as a power of two it folds into a single power. No big multiplications needed.

Step by step

Bring all the powers to one base: $4 = 2^2$, $8 = 2^3$, $\frac{1}{2} = 2^{-1}$.
When multiplying powers add the exponents, when dividing subtract them, when raising a power to a power multiply them.
Work out the final exponent and compute the power.

The exponent of the first power.

The exponent of the second power. The rules hold only when the bases are equal.

Example: $\frac{2^5 \cdot 4^3}{8^3} = \frac{2^5 \cdot 2^6}{2^9} = 2^{5 + 6 - 9} = 2^2 = 4$.

Common mistakes

Multiplying the exponents when multiplying powers: $2^3 \cdot 2^4 = 2^7$, not $2^{12}$.
Merging powers with different bases: $2^3 \cdot 3^2$ is not a single power.
Confusing a negative exponent with a negative number: $2^{-3} = \frac{1}{8}$, not $-8$.