Ctrl+k

Apply the product rule for exponents

The product rule for exponents expresses that multiplying powers with the same nonzero base combines their repeated factors, so am⋅an=am+na^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.

Detailed Explanation: Apply the product rule for exponents

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

am⋅an=am+na^m\cdot a^n=a^{m+n}

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

Example:

(3×104)(2×10−2)\left(3\times10^4\right)\left(2\times10^{-2}\right)

Step 1: Multiply the coefficients.

3⋅2=63\cdot2=6

Step 2: Use the product rule on the powers of 1010.

104⋅10−2=104+(−2)=10210^4\cdot10^{-2}=10^{4+(-2)}=10^2

Step 3: Combine the results.

(3×104)(2×10−2)=6×102\left(3\times10^4\right)\left(2\times10^{-2}\right)=6\times10^2

So, the answer is

6×102\boxed{6\times10^2}

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

Learn by doing: Apply the product rule for exponents

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 - Power Law with Composite Base (Negatives, Fraction with Power to Exponent)

      ?