If the base of the power is a product, as is the case with [[$ (4 \cdot 3)^2 $]], the expression is called a power of a product. These expressions be calculated using normal calculation rules: [[$ (4 \cdot 3)^2 = (12)^2 = 144 $]]. However, the power of a product also has its own calculation rules, which lead to the same result: [[$ (4 \cdot 3)^2 = 4^2 \cdot 3^2 = 16 \cdot 9 = 144 $]]

The power of a product

The power of a product is a product of the power of the factors.

[[$ (a\cdot b)^n = a^n \cdot b^n $]].

The rule for calculating the power of a product does not necessarily have to be used when calculating with numerical values alone. However, expressions that include variables cannot be simplified according to normal calculation rules.

Examples

Example 1

Simplify the powers.

a) [[$ (2 \cdot 3)^3 = 2^3 \cdot 3^3 = 8 \cdot 27 = 216 $]]

b) [[$ (2x)^3 = 2^3 \cdot x^3 = 8x^3 $]]

c) [[$ (5a^3b)^2 = 5^2 \cdot a^{3\cdot2} \cdot b^2 = 25a^6b^2 $]]

d) [[$ \dfrac{(2xy)^3}{2x^2} = \dfrac{2^3x^3y^3}{2x^2} = 2^{3-1} x^{3-2} y^3 = 2^2 xy^3 = 4xy^3 $]]