If `(4/5)^(n) = sqrt((5/4)^(3))`, then n = ?

Options

A. `-3/2`
B. `-1`
C. `-2/3`
D. `2/3`

Answer

To solve the equation \((\frac{4}{5})^{n} = \sqrt{(\frac{5}{4})^{3}}\), we will follow these steps:

### Step 1: Rewrite the square root as an exponent
The square root can be expressed as an exponent of \(\frac{1}{2}\). Thus, we can rewrite the right side of the equation:
\[ \sqrt{(\frac{5}{4})^{3}} = \left(\frac{5}{4}\right)^{3 \cdot \frac{1}{2}} = \left(\frac{5}{4}\right)^{\frac{3}{2}} \]

### Step 2: Rewrite the fraction \(\frac{5}{4}\)
We know that \(\frac{5}{4}\) can be rewritten as \(\left(\frac{4}{5}\right)^{-1}\). Therefore, we can express the right side as:
\[ \left(\frac{5}{4}\right)^{\frac{3}{2}} = \left(\left(\frac{4}{5}\right)^{-1}\right)^{\frac{3}{2}} = \left(\frac{4}{5}\right)^{-\frac{3}{2}} \]

### Step 3: Set the bases equal
Now we have:
\[ \left(\frac{4}{5}\right)^{n} = \left(\frac{4}{5}\right)^{-\frac{3}{2}} \]
Since the bases are the same, we can set the exponents equal to each other:
\[ n = -\frac{3}{2} \]

### Final Answer
Thus, the value of \(n\) is:
\[ n = -\frac{3}{2} \]