Skill: 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:

$$(10^3)^2$$

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

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

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

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

3. 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 $10^3\cdot10^2$.