`(64)^(-(2)/(3))xx((1)/(4))^(-2)` is equal to :

Options

A. `1`
B. `2`
C. `(1)/(2)`
D. `(1)/(16)`

Answer

To solve the expression \((64)^{-\frac{2}{3}} \times \left(\frac{1}{4}\right)^{-2}\), we can follow these steps:

### Step 1: Rewrite the expression using the property of negative exponents
Using the property that \(a^{-n} = \frac{1}{a^n}\), we can rewrite the expression:
\[ (64)^{-\frac{2}{3}} = \frac{1}{(64)^{\frac{2}{3}}} \]
Thus, the expression becomes:
\[ \frac{1}{(64)^{\frac{2}{3}}} \times \left(\frac{1}{4}\right)^{-2} \]

### Step 2: Simplify \(\left(\frac{1}{4}\right)^{-2}\)
Using the property of negative exponents again:
\[ \left(\frac{1}{4}\right)^{-2} = 4^2 = 16 \]
Now, our expression is:
\[ \frac{1}{(64)^{\frac{2}{3}}} \times 16 \]

### Step 3: Calculate \((64)^{\frac{2}{3}}\)
First, we find the cube root of 64. Since \(64 = 4^3\), we have:
\[ (64)^{\frac{1}{3}} = 4 \]
Now, squaring this result gives:
\[ (64)^{\frac{2}{3}} = (4)^2 = 16 \]

### Step 4: Substitute back into the expression
Now we substitute back into our expression:
\[ \frac{1}{(64)^{\frac{2}{3}}} \times 16 = \frac{1}{16} \times 16 \]

### Step 5: Simplify the expression
Finally, simplifying gives:
\[ \frac{1 \times 16}{16} = 1 \]

Thus, the final answer is:
\[ \boxed{1} \]

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