`(33^(2)-31^(2))^((6)/(7))`=____

Options

A. 64
B. 16
C. 32
D. 4

Answer

To solve the expression \((33^{2} - 31^{2})^{\frac{6}{7}}\), we will use the difference of squares identity. Here’s the step-by-step solution:

### Step 1: Apply the Difference of Squares Identity
The difference of squares identity states that \(a^{2} - b^{2} = (a - b)(a + b)\).
In our case, let \(a = 33\) and \(b = 31\).

\[ 33^{2} - 31^{2} = (33 - 31)(33 + 31) \]

### Step 2: Calculate \(33 - 31\) and \(33 + 31\)
Now we will calculate the values:

\[ 33 - 31 = 2 \]
\[ 33 + 31 = 64 \]

### Step 3: Substitute Back into the Expression
Now substitute these values back into the expression:

\[ 33^{2} - 31^{2} = 2 \times 64 \]

### Step 4: Calculate \(2 \times 64\)
Now we calculate:

\[ 2 \times 64 = 128 \]

### Step 5: Raise to the Power of \(\frac{6}{7}\)
Now we substitute this back into the original expression:

\[ (33^{2} - 31^{2})^{\frac{6}{7}} = 128^{\frac{6}{7}} \]

### Step 6: Simplify \(128^{\frac{6}{7}}\)
We know that \(128 = 2^{7}\), so we can rewrite it as:

\[ 128^{\frac{6}{7}} = (2^{7})^{\frac{6}{7}} = 2^{7 \cdot \frac{6}{7}} = 2^{6} \]

### Step 7: Calculate \(2^{6}\)
Finally, we calculate \(2^{6}\):

\[ 2^{6} = 64 \]

### Final Answer
Thus, the value of \((33^{2} - 31^{2})^{\frac{6}{7}}\) is:

\[ \boxed{64} \]