How do you turn 3^8 / 27^(-2) into 3^8 / 3^(-6), and what does the expression simplify to?

Step-by-step solution

1. 1

   Find the common base. Exponent rules only combine powers of the same base, so the first job is to express $27$ as a power of $3$:

   $27=3\times 3\times 3=3^{3}$

2. 2

   Substitute and apply the power-of-a-power rule. Replacing $27$ by $3^3$ and multiplying the exponents,

   $27^{-2}=\left(3^{3}\right)^{-2}=3^{3\times(-2)}=3^{-6}$

   This is the whole trick: $(a^m)^n=a^{mn}$, and the negative sign simply rides along in the product. That gives exactly the target form $\dfrac{3^{8}}{3^{-6}}$.

3. 3

   Divide the powers by subtracting exponents.

   $\frac{3^{8}}{3^{-6}}=3^{8-(-6)}=3^{14}$

   Subtracting a negative exponent adds, which is why dividing by $27^{-2}$ makes the expression larger.

4. 4

   Evaluate.

   $3^{14}=4\,782\,969$

5. 5

   Check without rewriting. Directly, $27^{-2}=1/27^2=1/729$, so $\dfrac{3^8}{1/729}=6561\times 729=4\,782\,969$ — the same value, confirming both the rewrite and the arithmetic.