Apply the power-of-a-product rule

The power of a product rule expresses that an exponent applies to every factor: $(ab)^n=a^n b^n$ , for numerical or algebraic factors and integer exponents where the expressions are defined. This structure supports efficient simplification of products, including scientific notation, and prevents the common error of applying the exponent to only one factor or adding exponents across multiplication; extensions involving rational exponents are not included here.

Detailed Explanation: Apply the power-of-a-product rule

When a product is raised to a power, the exponent applies to every factor inside the parentheses:

(ab)^n=a^n b^n

Example

Simplify:

(2x^3y)^4

Step 1: Apply the exponent to each factor.

The factors inside the parentheses are $2$ , $x^3$ , and $y$ :

(2x^3y)^4=2^4(x^3)^4y^4

Step 2: Simplify each power.

For a power raised to another power, multiply the exponents:

2^4=16,\qquad (x^3)^4=x^{3\cdot4}=x^{12}

So,

(2x^3y)^4=16x^{12}y^4

The exponent $4$ must be applied to every factor. Do not apply it only to $x$ , and do not add exponents across multiplication.

Learn by doing: Apply the power-of-a-product rule

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Exponents - Power of a Product with Variable Base (Product to Separate)

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