`sqrt(1 (1)/(4) xx (64)/(125) xx 1.44)` is equal to

Options

A. `1 (1)/(15)`
B. `(24)/(25)`
C. `(23)/(25)`
D. `(21)/(25)`

Answer

To solve the expression \( \sqrt{1 \cdot \frac{1}{4} \cdot \frac{64}{125} \cdot 1.44} \), we can follow these steps:

### Step 1: Simplify the expression inside the square root
We start with the expression:
\[ \sqrt{1 \cdot \frac{1}{4} \cdot \frac{64}{125} \cdot 1.44} \]
Since multiplying by 1 does not change the value, we can ignore it:
\[ \sqrt{\frac{1}{4} \cdot \frac{64}{125} \cdot 1.44} \]

### Step 2: Convert 1.44 to a fraction
Next, we convert 1.44 to a fraction. We know that:
\[ 1.44 = \frac{144}{100} \]
So we can rewrite the expression as:
\[ \sqrt{\frac{1}{4} \cdot \frac{64}{125} \cdot \frac{144}{100}} \]

### Step 3: Combine the fractions
Now, we can combine the fractions:
\[ \sqrt{\frac{1 \cdot 64 \cdot 144}{4 \cdot 125 \cdot 100}} \]

### Step 4: Simplify the numerator and denominator
Calculating the numerator:
\[ 1 \cdot 64 \cdot 144 = 64 \cdot 144 \]
Calculating the denominator:
\[ 4 \cdot 125 \cdot 100 = 50000 \]
Thus, we have:
\[ \sqrt{\frac{64 \cdot 144}{50000}} \]

### Step 5: Calculate the square root
Now we can simplify further. We know that:
\[ 64 = 8^2 \quad \text{and} \quad 144 = 12^2 \]
So:
\[ \sqrt{64 \cdot 144} = \sqrt{(8 \cdot 12)^2} = 8 \cdot 12 = 96 \]
And:
\[ \sqrt{50000} = \sqrt{500 \cdot 100} = \sqrt{500} \cdot 10 \]
Now, we can simplify \( \sqrt{500} \):
\[ \sqrt{500} = \sqrt{100 \cdot 5} = 10\sqrt{5} \]
Thus:
\[ \sqrt{50000} = 10 \cdot 10\sqrt{5} = 100\sqrt{5} \]

### Step 6: Final simplification
Now we have:
\[ \sqrt{\frac{64 \cdot 144}{50000}} = \frac{96}{100\sqrt{5}} = \frac{96}{100\sqrt{5}} = \frac{24}{25\sqrt{5}} \]

### Final Answer
Thus, the final answer is:
\[ \frac{24}{25} \]