Skill: Apply the product rule for powers

Explanation and Free Practice Resources

For powers with the same base, multiplying $a^m$ by $a^n$ combines the repeated factors, giving $a^{m+n}$ ; the exponent records the total number of factors of $a$ , rather than being multiplied. This rule applies to numerical or algebraic expressions with whole-number exponents and supports efficient computation and scientific notation, while unlike bases and extensions to negative or fractional exponents are not included.

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Detailed Explanation: Apply the product rule for powers

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

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

The exponents are added because they count the total number of factors of $a$ .

Example

Simplify:

2^3\cdot 2^4

Step 1: Check the bases.
Both powers have base $2$ , so the product rule applies.

Step 2: Add the exponents.

2^3\cdot 2^4=2^{3+4}

Step 3: Simplify the exponent.

2^{3+4}=2^7

So,

\boxed{2^3\cdot 2^4=2^7}

The exponents are added, not multiplied, because there are $3+4=7$ total factors of $2$ :

(2\cdot2\cdot2)(2\cdot2\cdot2\cdot2)=2^7

Learn by doing: Apply the product rule for powers

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Exponents - Multiplication - Positive by Positive to Positive

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