If `(16^(2) - 11^(2))^(2//3) = 3^(2) xx A`, then the value of A is

Options

A. `3sqrt(25)`
B. `3sqrt(75)`
C. 25
D. 5

Answer

To solve the equation \((16^2 - 11^2)^{\frac{2}{3}} = 3^2 \times A\), we will follow these steps:

### Step 1: Calculate \(16^2\) and \(11^2\)
First, we need to calculate the squares of 16 and 11.

\[ 16^2 = 256 \]
\[ 11^2 = 121 \]

### Step 2: Subtract the squares
Now, we will subtract \(11^2\) from \(16^2\).

\[ 16^2 - 11^2 = 256 - 121 = 135 \]

### Step 3: Raise the result to the power of \(\frac{2}{3}\)
Next, we raise the result to the power of \(\frac{2}{3}\).

\[ (135)^{\frac{2}{3}} \]

### Step 4: Simplify \(135\)
We can express 135 in terms of its prime factors:

\[ 135 = 3^3 \times 5 \]

### Step 5: Apply the power to the factors
Now we can apply the power of \(\frac{2}{3}\) to the prime factorization:

\[ (3^3 \times 5)^{\frac{2}{3}} = (3^3)^{\frac{2}{3}} \times (5)^{\frac{2}{3}} = 3^{3 \times \frac{2}{3}} \times 5^{\frac{2}{3}} = 3^2 \times 5^{\frac{2}{3}} \]

### Step 6: Set the equation equal to \(3^2 \times A\)
Now we can set our result equal to \(3^2 \times A\):

\[ 3^2 \times 5^{\frac{2}{3}} = 3^2 \times A \]

### Step 7: Cancel \(3^2\) from both sides
We can cancel \(3^2\) from both sides of the equation:

\[ 5^{\frac{2}{3}} = A \]

### Step 8: Final value of \(A\)
Thus, the value of \(A\) is:

\[ A = 5^{\frac{2}{3}} \]