High School

Which of the following is equal to the fraction below?

[tex]\left(\frac{4}{5}\right)^6[/tex]

A. [tex]\frac{4^6}{5}[/tex]

B. [tex]\frac{4^6}{5^6}[/tex]

C. [tex]6 \cdot \left(\frac{4}{5}\right)[/tex]

D. [tex]\frac{24}{30}[/tex]

Answer :

To solve the problem, we start with the expression

$$\left(\frac{4}{5}\right)^6.$$

When raising a fraction to a power, we apply the exponent to both the numerator and the denominator. This follows the exponent rule

$$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}.$$

Applying this rule, we have:

$$\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}.$$

Now, let's compare this with the provided options:

- Option A: $\frac{4^6}{5}$
The denominator is not raised to the power 6, so this is not equivalent.

- Option B: $\frac{4^6}{5^6}$
This exactly matches our expression.

- Option C: $6 \cdot \left(\frac{4}{5}\right)$
Multiplying by 6 does not give the same result as raising the fraction to the 6th power.

- Option D: $\frac{24}{30}$
This simplifies to $\frac{4}{5}$, which is not equal to $\left(\frac{4}{5}\right)^6$.

Thus, the expression

$$\left(\frac{4}{5}\right)^6$$

is equal to

$$\frac{4^6}{5^6},$$

which corresponds to option B.