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)/(sqrt(2))`
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
First, we convert the mixed numbers \(3 \frac{1}{4}\) 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 Values into the Expression
Now, we 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
We can use the difference of squares formula, which states that \(x^2 - y^2 = (x - y)(x + y)\). Thus, we can rewrite the numerator:

\[ a^4 - b^4 = (a^2 - b^2)(a^2 + b^2) \]

Substituting this back into the expression gives:

\[ \sqrt{\frac{(a^2 - b^2)(a^2 + b^2)}{a^2 - b^2}} \]

### Step 4: Simplify the Expression
The \(a^2 - b^2\) terms in the numerator and denominator cancel out:

\[ \sqrt{a^2 + b^2} \]

### Step 5: Calculate \(a^2\) and \(b^2\)
Now we need to 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 6: 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 7: Add \(a^2\) and \(b^2\)
Now we can add \(a^2\) and \(b^2\):

\[ a^2 + b^2 = \frac{1521}{144} + \frac{2704}{144} = \frac{1521 + 2704}{144} = \frac{4225}{144} \]

### Step 8: Take the Square Root
Now we take the square root of the sum:

\[ \sqrt{a^2 + b^2} = \sqrt{\frac{4225}{144}} = \frac{\sqrt{4225}}{\sqrt{144}} = \frac{65}{12} \]

### Final Result
Thus, the final result is:

\[ \frac{65}{12} \]