What is the simplified from of `sqrt((64)/(288)) `?

Options

A. `(sqrt(2))/(3)`
B. `(1)/(2)`
C. `(2)/(sqrt(3))`
D. `(3)/(sqrt(3))`

Answer

To simplify the expression \(\sqrt{\frac{64}{288}}\), we can follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ \sqrt{\frac{64}{288}} \]

### Step 2: Simplify the fraction inside the square root
We can simplify the fraction \(\frac{64}{288}\) by finding the greatest common divisor (GCD) of 64 and 288.

The prime factorization of 64 is:
\[ 64 = 2^6 \]
The prime factorization of 288 is:
\[ 288 = 2^5 \times 3^2 \]

The GCD of 64 and 288 is \(2^5 = 32\). Now we can divide both the numerator and the denominator by 32:
\[ \frac{64 \div 32}{288 \div 32} = \frac{2}{9} \]

### Step 3: Substitute back into the square root
Now we substitute this simplified fraction back into the square root:
\[ \sqrt{\frac{2}{9}} \]

### Step 4: Simplify the square root
We can simplify the square root of a fraction:
\[ \sqrt{\frac{2}{9}} = \frac{\sqrt{2}}{\sqrt{9}} = \frac{\sqrt{2}}{3} \]

### Final Answer
Thus, the simplified form of \(\sqrt{\frac{64}{288}}\) is:
\[ \frac{\sqrt{2}}{3} \]

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