Skill: Apply the product rule for exponents

Explanation and Free Practice Resources

The product rule for exponents expresses that multiplying powers with the same nonzero base combines their repeated factors, so $a^m\cdot a^n=a^{m+n}$ for integer exponents, including zero and negative exponents when defined. The coefficient and base must be handled separately in expressions such as scientific notation, and exponents are added—not multiplied; this scope does not extend to arbitrary real or complex exponents.

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Detailed Explanation: Apply the product rule for exponents

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

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

The exponents are added, not multiplied. If there are coefficients, multiply the coefficients separately.

Example:

\left(3\times10^4\right)\left(2\times10^{-2}\right)

Step 1: Multiply the coefficients.

3\cdot2=6

Step 2: Use the product rule on the powers of $10$ .

10^4\cdot10^{-2}=10^{4+(-2)}=10^2

Step 3: Combine the results.

\left(3\times10^4\right)\left(2\times10^{-2}\right)=6\times10^2

So, the answer is

\boxed{6\times10^2}

Remember: the bases must be the same, and you add their exponents.

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Exponents - Multiplication - Positive by Positive to Positive

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