`sqrt(1 9/16)-sqrt(1 7/9)=`______

Options

A. 44208
B. 44230
C. `-1/12`
D. `-2//3`

Answer

To solve the expression \( \sqrt{1 \frac{9}{16}} - \sqrt{1 \frac{7}{9}} \), 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 \( 1 \frac{9}{16} \):
\[ 1 \frac{9}{16} = \frac{1 \times 16 + 9}{16} = \frac{16 + 9}{16} = \frac{25}{16} \]

- 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:
\[ \sqrt{\frac{25}{16}} - \sqrt{\frac{16}{9}} \]

### Step 3: Calculate the Square Roots
Next, we calculate the square roots of the fractions:
- \( \sqrt{\frac{25}{16}} = \frac{\sqrt{25}}{\sqrt{16}} = \frac{5}{4} \)
- \( \sqrt{\frac{16}{9}} = \frac{\sqrt{16}}{\sqrt{9}} = \frac{4}{3} \)

### Step 4: Rewrite the Expression Again
Now we can rewrite the expression with the calculated square roots:
\[ \frac{5}{4} - \frac{4}{3} \]

### Step 5: Find a Common Denominator
To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 4 and 3 is 12.

- Convert \( \frac{5}{4} \) to have a denominator of 12:
\[ \frac{5}{4} = \frac{5 \times 3}{4 \times 3} = \frac{15}{12} \]

- Convert \( \frac{4}{3} \) to have a denominator of 12:
\[ \frac{4}{3} = \frac{4 \times 4}{3 \times 4} = \frac{16}{12} \]

### Step 6: Subtract the Fractions
Now we can subtract the two fractions:
\[ \frac{15}{12} - \frac{16}{12} = \frac{15 - 16}{12} = \frac{-1}{12} \]

### Final Answer
Thus, the final answer is:
\[ \sqrt{1 \frac{9}{16}} - \sqrt{1 \frac{7}{9}} = -\frac{1}{12} \]

---