Simplify the Expression: (6⁸ × 7⁸) ÷ 17⁸

The product rule for exponents only works when the exponents are identical! $a^n \times b^n = (a \times b)^n$, but if exponents differ, you cannot combine them this way.

Not necessarily! The question asks for the simplified expression, which is $\frac{(6 \times 7)^8}{17^8}$. You could calculate $\frac{42^8}{17^8}$, but that's not required for this problem.

No! You can only cancel identical factors. Since $(6 \times 7)^8 \neq 17^8$, the powers don't cancel. The expression $\frac{(6 \times 7)^8}{17^8}$ is already in simplest form.

If exponents are different, like $6^5 \times 7^8$, you cannot use the product rule. Each term must have the same exponent to combine using $a^n \times b^n = (a \times b)^n$.

Yes! You could also write it as $\left(\frac{6 \times 7}{17}\right)^8$ or $\left(\frac{42}{17}\right)^8$, but the form $\frac{(6 \times 7)^8}{17^8}$ clearly shows the product rule application.