The value of `(sqrt(80) - sqrt(112))/(sqrt(45) - sqrt(63))` is :

Options

A. `(3)/(4)`
B. `1 (3)/(4)`
C. `1 (1)/(3)`
D. `1 (7)/(9)`

Answer

To solve the expression \((\sqrt{80} - \sqrt{112}) / (\sqrt{45} - \sqrt{63})\), we will simplify both the numerator and the denominator step by step.

### Step 1: Simplify \(\sqrt{80}\) and \(\sqrt{112}\)

- \(\sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5}\)
- \(\sqrt{112} = \sqrt{16 \times 7} = \sqrt{16} \times \sqrt{7} = 4\sqrt{7}\)

Now, substituting these values into the numerator:

\[ \sqrt{80} - \sqrt{112} = 4\sqrt{5} - 4\sqrt{7} = 4(\sqrt{5} - \sqrt{7}) \]

### Step 2: Simplify \(\sqrt{45}\) and \(\sqrt{63}\)

- \(\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}\)
- \(\sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} = 3\sqrt{7}\)

Now, substituting these values into the denominator:

\[ \sqrt{45} - \sqrt{63} = 3\sqrt{5} - 3\sqrt{7} = 3(\sqrt{5} - \sqrt{7}) \]

### Step 3: Substitute back into the original expression

Now we can substitute the simplified numerator and denominator back into the expression:

\[ \frac{\sqrt{80} - \sqrt{112}}{\sqrt{45} - \sqrt{63}} = \frac{4(\sqrt{5} - \sqrt{7})}{3(\sqrt{5} - \sqrt{7})} \]

### Step 4: Cancel out the common terms

Since \((\sqrt{5} - \sqrt{7})\) is common in both the numerator and the denominator, we can cancel it out (assuming \(\sqrt{5} \neq \sqrt{7}\)):

\[ = \frac{4}{3} \]

### Final Result

Thus, the value of the expression \((\sqrt{80} - \sqrt{112}) / (\sqrt{45} - \sqrt{63})\) is:

\[ \frac{4}{3} \]