Apply the power-of-a-power rule

A nested power such as $(a^m)^n$ represents raising a quantity already formed by repeated multiplication to another whole-number power, so its equivalent form is $a^{mn}$ : the exponents are multiplied, not added. This reasoning applies to numerical bases, including powers of 10 used in scientific notation, and helps preserve the distinction from the product-of-powers rule; fractional exponents and more advanced generalizations are outside this scope.

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

When a power is raised to another power, multiply the exponents:

(a^m)^n=a^{m\cdot n}

For example, simplify:

(10^3)^2

The outside exponent, $2$ , means that $10^3$ is used as a factor twice:

(10^3)^2=10^3\cdot10^3

Because the bases are both $10$ , use the product-of-powers rule to add the exponents:

10^3\cdot10^3=10^{3+3}=10^6

Therefore:

\boxed{(10^3)^2=10^6}

The exponents were multiplied because this was a power raised to a power:

3\cdot2=6

Do not add the exponents directly in $(10^3)^2$ ; adding is used when multiplying powers with the same base, such as $10^3\cdot10^2$ .

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

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Exponents - Power Law with Variable Base (Positives, Exponent with Power to Exponent)

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Apply the power-of-a-power rule

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

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

Practice with unlimited practice problems

Practice:

Exponents - Power Law with Variable Base (Positives, Exponent with Power to Exponent)