The square root of `((3 (1)/(4))^(4) - (4 (1)/(3))^(4))/((3 (1)/(4))^(2) - (4 (1)/(3))^(2))` is :

Options

A. `7 (1)/(12)`
B. `5 (5)/(12)`
C. `1 (1)/(12)`
D. `1 (7)/(12)`

Answer

To solve the expression \(\sqrt{\frac{(3 \frac{1}{4})^4 - (4 \frac{1}{3})^4}{(3 \frac{1}{4})^2 - (4 \frac{1}{3})^2}}\), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
Convert \(3 \frac{1}{4}\) and \(4 \frac{1}{3}\) into improper fractions.
- \(3 \frac{1}{4} = \frac{3 \times 4 + 1}{4} = \frac{12 + 1}{4} = \frac{13}{4}\)
- \(4 \frac{1}{3} = \frac{4 \times 3 + 1}{3} = \frac{12 + 1}{3} = \frac{13}{3}\)

### Step 2: Substitute the Values
Now substitute \(A = \frac{13}{4}\) and \(B = \frac{13}{3}\) into the expression:
\[ \sqrt{\frac{A^4 - B^4}{A^2 - B^2}} \]

### Step 3: Apply the Difference of Squares Formula
Recall that \(A^4 - B^4 = (A^2 - B^2)(A^2 + B^2)\). Therefore, we can simplify:
\[ \frac{A^4 - B^4}{A^2 - B^2} = A^2 + B^2 \]

### Step 4: Calculate \(A^2\) and \(B^2\)
Now calculate \(A^2\) and \(B^2\):
- \(A^2 = \left(\frac{13}{4}\right)^2 = \frac{169}{16}\)
- \(B^2 = \left(\frac{13}{3}\right)^2 = \frac{169}{9}\)

### Step 5: Find a Common Denominator
To add \(A^2\) and \(B^2\), we need a common denominator. The least common multiple of 16 and 9 is 144.
- Convert \(A^2\):
\[ A^2 = \frac{169}{16} = \frac{169 \times 9}{16 \times 9} = \frac{1521}{144} \]
- Convert \(B^2\):
\[ B^2 = \frac{169}{9} = \frac{169 \times 16}{9 \times 16} = \frac{2704}{144} \]

### Step 6: Add \(A^2\) and \(B^2\)
Now add the two fractions:
\[ A^2 + B^2 = \frac{1521}{144} + \frac{2704}{144} = \frac{1521 + 2704}{144} = \frac{4225}{144} \]

### Step 7: Take the Square Root
Now we take the square root of the result:
\[ \sqrt{A^2 + B^2} = \sqrt{\frac{4225}{144}} = \frac{\sqrt{4225}}{\sqrt{144}} = \frac{65}{12} \]

### Final Answer
Thus, the final answer is:
\[ \frac{65}{12} \]

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