Apply exponent laws for products

Exponent laws for products express repeated multiplication compactly: for a common nonzero base, $a^m\cdot a^n=a^{m+n}$ , and for a product raised to a positive integer power, $(ab)^n=a^n b^n$ . The reasoning distinguishes multiplying powers from raising a power to a power—exponents are added in the former case, not multiplied or applied to the bases—and supports simplifying algebraic expressions; fractional or negative exponents and more advanced edge cases are outside this scope.

Detailed Explanation: Apply exponent laws for products

When multiplying powers with the same base, add the exponents:

a^m\cdot a^n=a^{m+n}

When a product is raised to a power, apply the exponent to each factor:

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

Worked example

Simplify:

x^3\cdot x^2\cdot(2x)^2

Step 1: Multiply the powers with the same base.

The first two factors both have base $x$ , so add their exponents:

x^3\cdot x^2=x^{3+2}=x^5

Now the expression is

x^5\cdot(2x)^2

Step 2: Apply the exponent to each factor inside the parentheses.

(2x)^2=2^2x^2=4x^2

So the expression becomes

x^5\cdot4x^2

Step 3: Multiply the powers with base $x$ .

Add the exponents:

x^5\cdot4x^2=4x^{5+2}=4x^7

Therefore,

\boxed{4x^7}

Remember: when multiplying powers with the same base, add the exponents. When raising a product to a power, apply the exponent to every factor.

Learn by doing: Apply exponent laws for products

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Practice with unlimited practice problems

Exponents - Multiplication - Positive by Positive to Positive

Earned ?