Write the fraction 256/625 as a power.

Step-by-step solution

1. 1

   Factor each part into a prime power first. Repeated division shows

   $256=2^{8},\qquad 625=5^{4}$

   This already answers the question in one form, $\dfrac{2^{8}}{5^{4}}$, but the exponents differ, so the fraction cannot yet be written as a single power.

2. 2

   Look for a common exponent. Since $8=4\cdot 2$, regroup the numerator as a fourth power:

   $2^{8}=\left(2^{2}\right)^{4}=4^{4}$

   Now both numerator and denominator carry the same exponent $4$: $256=4^4$ and $625=5^4$.

3. 3

   Fold the equal exponents into one power of a fraction. The quotient rule for powers, $\dfrac{a^{n}}{b^{n}}=\left(\dfrac{a}{b}\right)^{n}$, applies exactly when the exponents match:

   $\frac{256}{625}=\frac{4^{4}}{5^{4}}=\left(\frac{4}{5}\right)^{4}$

4. 4

   Check the value. $\left(\dfrac{4}{5}\right)^{4}=\dfrac{4\cdot 4\cdot 4\cdot 4}{5\cdot 5\cdot 5\cdot 5}=\dfrac{256}{625}$. ✓ Also note $\dfrac{256}{625}$ is already in lowest terms, since $2$ and $5$ share no factor — that is why no cancelling was possible.

5. 5

   Know which form to hand in. $\left(\dfrac{4}{5}\right)^{4}$ is the requested single power; $\dfrac{2^{8}}{5^{4}}$ is the prime-factored form. Both are correct, but only the first is one power of one base.