If `sqrt(21)= 4.58` , what is the simplified value of `(8sqrt((3)/(7))-3sqrt((7)/(3)))` ?

Options

A. a) `0.547`
B. b) `0.752`
C. c) `0.655`
D. d) `1`

Answer

To simplify the expression \(8\sqrt{\frac{3}{7}} - 3\sqrt{\frac{7}{3}}\), we can follow these steps:

### Step 1: Rewrite the square roots
We start by rewriting the square roots in the expression:
\[ 8\sqrt{\frac{3}{7}} - 3\sqrt{\frac{7}{3}} = 8\frac{\sqrt{3}}{\sqrt{7}} - 3\frac{\sqrt{7}}{\sqrt{3}} \]

### Step 2: Find a common denominator
To combine the terms, we need a common denominator, which is \(\sqrt{21}\) (since \(\sqrt{7} \cdot \sqrt{3} = \sqrt{21}\)):
\[ = \frac{8\sqrt{3} \cdot \sqrt{3}}{\sqrt{21}} - \frac{3\sqrt{7} \cdot \sqrt{7}}{\sqrt{21}} \]
This simplifies to:
\[ = \frac{8 \cdot 3 - 3 \cdot 7}{\sqrt{21}} \]

### Step 3: Simplify the numerator
Now we calculate the numerator:
\[ = \frac{24 - 21}{\sqrt{21}} = \frac{3}{\sqrt{21}} \]

### Step 4: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and the denominator by \(\sqrt{21}\):
\[ = \frac{3\sqrt{21}}{21} \]

### Step 5: Simplify the fraction
Now we simplify the fraction:
\[ = \frac{\sqrt{21}}{7} \]

### Step 6: Substitute the value of \(\sqrt{21}\)
Given that \(\sqrt{21} = 4.58\), we substitute this value into the expression:
\[ = \frac{4.58}{7} \]

### Step 7: Calculate the final value
Now we perform the division:
\[ = 0.6542857142857143 \approx 0.655 \]

Thus, the simplified value of \(8\sqrt{\frac{3}{7}} - 3\sqrt{\frac{7}{3}}\) is approximately \(0.655\).

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