Find the value of `( sqrt((625)/(4356))+ sqrt((576)/(1089))) xx ((66)/(sqrt(19600)+ sqrt(36)))`

Options

A. `(7)/(15)`
B. `(9)/(53)`
C. `(1)/(2)`
D. `(79)/(33)`

Answer

To solve the expression \( \left( \sqrt{\frac{625}{4356}} + \sqrt{\frac{576}{1089}} \right) \times \left( \frac{66}{\sqrt{19600} + \sqrt{36}} \right) \), we will break it down step by step.

### Step 1: Simplify \( \sqrt{\frac{625}{4356}} \)

1. Calculate \( \sqrt{625} \) and \( \sqrt{4356} \):
- \( \sqrt{625} = 25 \)
- \( \sqrt{4356} = 66 \) (since \( 66^2 = 4356 \))

2. Therefore, \( \sqrt{\frac{625}{4356}} = \frac{\sqrt{625}}{\sqrt{4356}} = \frac{25}{66} \).

### Step 2: Simplify \( \sqrt{\frac{576}{1089}} \)

1. Calculate \( \sqrt{576} \) and \( \sqrt{1089} \):
- \( \sqrt{576} = 24 \)
- \( \sqrt{1089} = 33 \) (since \( 33^2 = 1089 \))

2. Therefore, \( \sqrt{\frac{576}{1089}} = \frac{\sqrt{576}}{\sqrt{1089}} = \frac{24}{33} \).

### Step 3: Combine the results from Step 1 and Step 2

Now we add the two fractions:
\[ \frac{25}{66} + \frac{24}{33} \]

To add these fractions, we need a common denominator. The least common multiple of 66 and 33 is 66.

1. Convert \( \frac{24}{33} \) to have a denominator of 66:
\[ \frac{24}{33} = \frac{24 \times 2}{33 \times 2} = \frac{48}{66} \]

2. Now add the fractions:
\[ \frac{25}{66} + \frac{48}{66} = \frac{25 + 48}{66} = \frac{73}{66} \]

### Step 4: Simplify \( \sqrt{19600} + \sqrt{36} \)

1. Calculate \( \sqrt{19600} \) and \( \sqrt{36} \):
- \( \sqrt{19600} = 140 \)
- \( \sqrt{36} = 6 \)

2. Therefore,
\[ \sqrt{19600} + \sqrt{36} = 140 + 6 = 146 \]

### Step 5: Calculate the final expression

Now we substitute back into the expression:
\[ \left( \frac{73}{66} \right) \times \left( \frac{66}{146} \right) \]

1. The \( 66 \) in the numerator and denominator cancels out:
\[ \frac{73}{146} \]

2. Simplifying \( \frac{73}{146} \):
- \( 146 = 2 \times 73 \), so:
\[ \frac{73}{146} = \frac{73}{2 \times 73} = \frac{1}{2} \]

### Final Answer

Thus, the value of the expression is \( \frac{1}{2} \).

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