`(3(6)/(17)div2(7)/(34)-1(9)/(25))=(?)^(2)`

Options

A. `2/5`
B. `1/3`
C. `4/5`
D. 1/5`

Explanation

To solve the equation \((\frac{36}{17} \div \frac{27}{34} - \frac{19}{25}) = (?)^2\), we will follow these steps:

### Step 1: Rewrite the division as multiplication
We can rewrite the division of fractions as multiplication by the reciprocal. Therefore, we have:
\[ \frac{36}{17} \div \frac{27}{34} = \frac{36}{17} \times \frac{34}{27} \]

### Step 2: Multiply the fractions
Now we can multiply the fractions:
\[ \frac{36 \times 34}{17 \times 27} \]

### Step 3: Simplify the multiplication
Calculating the numerator and denominator:
- Numerator: \(36 \times 34 = 1224\)
- Denominator: \(17 \times 27 = 459\)

So we have:
\[ \frac{1224}{459} \]

### Step 4: Simplify the fraction
Now we simplify \(\frac{1224}{459}\). We can find the greatest common divisor (GCD) of 1224 and 459. The GCD is 51. Dividing both the numerator and denominator by 51 gives:
\[ \frac{1224 \div 51}{459 \div 51} = \frac{24}{9} = \frac{8}{3} \]

### Step 5: Subtract the second fraction
Now we need to subtract \(\frac{19}{25}\) from \(\frac{8}{3}\). To do this, we need a common denominator. The least common multiple (LCM) of 3 and 25 is 75. We convert both fractions:
\[ \frac{8}{3} = \frac{8 \times 25}{3 \times 25} = \frac{200}{75} \]
\[ \frac{19}{25} = \frac{19 \times 3}{25 \times 3} = \frac{57}{75} \]

Now we can subtract:
\[ \frac{200}{75} - \frac{57}{75} = \frac{200 - 57}{75} = \frac{143}{75} \]

### Step 6: Set up the equation
Now we have:
\[ \frac{143}{75} = x^2 \]

### Step 7: Solve for x
To find \(x\), we take the square root of both sides:
\[ x = \sqrt{\frac{143}{75}} = \frac{\sqrt{143}}{\sqrt{75}} = \frac{\sqrt{143}}{5\sqrt{3}} = \frac{\sqrt{143}}{5\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{429}}{15} \]

### Final Answer
Thus, the value of \(x\) is:
\[ x = \frac{\sqrt{429}}{15} \]