Find the value of `sqrt(20 1/4)-sqrt(1 32/49)`.

Answer

To solve the expression \( \sqrt{20 \frac{1}{4}} - \sqrt{1 \frac{32}{49}} \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert the mixed numbers into improper fractions.

For \( 20 \frac{1}{4} \):
\[ 20 \frac{1}{4} = \frac{20 \times 4 + 1}{4} = \frac{80 + 1}{4} = \frac{81}{4} \]

For \( 1 \frac{32}{49} \):
\[ 1 \frac{32}{49} = \frac{1 \times 49 + 32}{49} = \frac{49 + 32}{49} = \frac{81}{49} \]

### Step 2: Rewrite the Expression
Now we can rewrite the original expression using the improper fractions:
\[ \sqrt{20 \frac{1}{4}} - \sqrt{1 \frac{32}{49}} = \sqrt{\frac{81}{4}} - \sqrt{\frac{81}{49}} \]

### Step 3: Simplify the Square Roots
Next, we simplify the square roots:
\[ \sqrt{\frac{81}{4}} = \frac{\sqrt{81}}{\sqrt{4}} = \frac{9}{2} \]
\[ \sqrt{\frac{81}{49}} = \frac{\sqrt{81}}{\sqrt{49}} = \frac{9}{7} \]

### Step 4: Subtract the Two Values
Now we can subtract the two fractions:
\[ \frac{9}{2} - \frac{9}{7} \]

To subtract these fractions, we need a common denominator. The least common multiple of 2 and 7 is 14.

Convert each fraction:
\[ \frac{9}{2} = \frac{9 \times 7}{2 \times 7} = \frac{63}{14} \]
\[ \frac{9}{7} = \frac{9 \times 2}{7 \times 2} = \frac{18}{14} \]

Now, subtract:
\[ \frac{63}{14} - \frac{18}{14} = \frac{63 - 18}{14} = \frac{45}{14} \]

### Final Answer
Thus, the value of \( \sqrt{20 \frac{1}{4}} - \sqrt{1 \frac{32}{49}} \) is:
\[ \frac{45}{14} \]

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