Simplify: `sqrt(2 14/25)-sqrt(1 7/9)`

Answer

To simplify the expression \( \sqrt{2 \frac{14}{25}} - \sqrt{1 \frac{7}{9}} \), we will follow these steps:

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

1. For \( 2 \frac{14}{25} \):
\[ 2 \frac{14}{25} = \frac{2 \times 25 + 14}{25} = \frac{50 + 14}{25} = \frac{64}{25} \]

2. For \( 1 \frac{7}{9} \):
\[ 1 \frac{7}{9} = \frac{1 \times 9 + 7}{9} = \frac{9 + 7}{9} = \frac{16}{9} \]

### Step 2: Rewrite the Expression
Now, we can rewrite the original expression using the improper fractions:
\[ \sqrt{\frac{64}{25}} - \sqrt{\frac{16}{9}} \]

### Step 3: Simplify Each Square Root
Next, we simplify each square root:

1. For \( \sqrt{\frac{64}{25}} \):
\[ \sqrt{\frac{64}{25}} = \frac{\sqrt{64}}{\sqrt{25}} = \frac{8}{5} \]

2. For \( \sqrt{\frac{16}{9}} \):
\[ \sqrt{\frac{16}{9}} = \frac{\sqrt{16}}{\sqrt{9}} = \frac{4}{3} \]

### Step 4: Subtract the Two Results
Now we subtract the two simplified square roots:
\[ \frac{8}{5} - \frac{4}{3} \]

### Step 5: Find a Common Denominator
To perform the subtraction, we need a common denominator. The least common multiple of 5 and 3 is 15.

1. Convert \( \frac{8}{5} \) to have a denominator of 15:
\[ \frac{8}{5} = \frac{8 \times 3}{5 \times 3} = \frac{24}{15} \]

2. Convert \( \frac{4}{3} \) to have a denominator of 15:
\[ \frac{4}{3} = \frac{4 \times 5}{3 \times 5} = \frac{20}{15} \]

### Step 6: Perform the Subtraction
Now we can subtract:
\[ \frac{24}{15} - \frac{20}{15} = \frac{24 - 20}{15} = \frac{4}{15} \]

### Final Answer
Thus, the simplified expression is:
\[ \frac{4}{15} \]

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