If `sqrt(2.5)=1.581`,then find the value of `sqrt(0.625)`.

Options

A. `0.7905`
B. `0.9426`
C. `0.7632`
D. `0.9325`

Explanation

To find the value of \( \sqrt{0.625} \) given that \( \sqrt{2.5} = 1.581 \), we can follow these steps:

### Step 1: Rewrite \( 0.625 \) as a fraction
We can express \( 0.625 \) as a fraction:
\[ 0.625 = \frac{625}{1000} \]

### Step 2: Simplify the fraction
Next, we can simplify \( \frac{625}{1000} \):
\[ \frac{625}{1000} = \frac{625 \div 125}{1000 \div 125} = \frac{5}{8} \]

### Step 3: Rewrite \( \sqrt{0.625} \)
Now we can rewrite \( \sqrt{0.625} \) using our simplified fraction:
\[ \sqrt{0.625} = \sqrt{\frac{5}{8}} = \frac{\sqrt{5}}{\sqrt{8}} \]

### Step 4: Simplify \( \sqrt{8} \)
We know that \( \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \). Thus,
\[ \sqrt{0.625} = \frac{\sqrt{5}}{2\sqrt{2}} \]

### Step 5: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and the denominator by \( \sqrt{2} \):
\[ \sqrt{0.625} = \frac{\sqrt{5} \cdot \sqrt{2}}{2\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{10}}{4} \]

### Step 6: Calculate \( \sqrt{10} \)
Next, we can express \( \sqrt{10} \) in terms of \( \sqrt{2.5} \):
\[ \sqrt{10} = \sqrt{4 \times 2.5} = 2\sqrt{2.5} \]
Thus,
\[ \sqrt{0.625} = \frac{2\sqrt{2.5}}{4} = \frac{\sqrt{2.5}}{2} \]

### Step 7: Substitute the value of \( \sqrt{2.5} \)
Now we can substitute the given value of \( \sqrt{2.5} \):
\[ \sqrt{0.625} = \frac{1.581}{2} \]

### Step 8: Calculate the final value
Now we perform the division:
\[ \sqrt{0.625} = 0.7905 \]

Thus, the value of \( \sqrt{0.625} \) is \( 0.7905 \).

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