the value of `(sqrt80 - sqrt112)/(sqrt45 - sqrt63)` 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 the square roots in the numerator
We can express \(\sqrt{80}\) and \(\sqrt{112}\) in terms of their prime factors:
- \(\sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5}\)
- \(\sqrt{112} = \sqrt{16 \times 7} = \sqrt{16} \times \sqrt{7} = 4\sqrt{7}\)

So, the numerator becomes:
\[ \sqrt{80} - \sqrt{112} = 4\sqrt{5} - 4\sqrt{7} = 4(\sqrt{5} - \sqrt{7}) \]

### Step 2: Simplify the square roots in the denominator
Now, we simplify \(\sqrt{45}\) and \(\sqrt{63}\):
- \(\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}\)
- \(\sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} = 3\sqrt{7}\)

So, the denominator becomes:
\[ \sqrt{45} - \sqrt{63} = 3\sqrt{5} - 3\sqrt{7} = 3(\sqrt{5} - \sqrt{7}) \]

### Step 3: Substitute back into the expression
Now we 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 the common terms
Since \(\sqrt{5} - \sqrt{7}\) is common in both the numerator and the denominator, we can cancel it out (assuming \(\sqrt{5} \neq \sqrt{7}\)):
\[ = \frac{4}{3} \]

### Step 5: Convert to mixed fraction (if needed)
The fraction \(\frac{4}{3}\) can be expressed as a mixed number:
\[ 4 \div 3 = 1 \quad \text{with a remainder of } 1 \]
Thus, it can be written as:
\[ 1 \frac{1}{3} \]

### Final Answer
The value of \((\sqrt{80} - \sqrt{112}) / (\sqrt{45} - \sqrt{63})\) is \(\frac{4}{3}\) or \(1 \frac{1}{3}\).