What is the square root of `(0.324xx0.64xx129.6)/(0.729xx1.024xx36`

Options

A. 4
B. 3
C. 2
D. 1

Answer

To solve the problem, we need to find the square root of the expression:

\[ \frac{0.324 \times 0.64 \times 129.6}{0.729 \times 1.024 \times 36} \]

### Step 1: Remove the decimals
To simplify the calculations, we can eliminate the decimals by multiplying both the numerator and the denominator by \(10^6\) (since the largest number of decimal places is 3).

This gives us:

\[ \frac{0.324 \times 10^6 \times 0.64 \times 10^6 \times 129.6 \times 10^6}{0.729 \times 10^6 \times 1.024 \times 10^6 \times 36 \times 10^6} \]

Calculating the new values:

\[ = \frac{324 \times 640 \times 129600}{729 \times 1024 \times 3600} \]

### Step 2: Simplify the expression
Now we can simplify the expression step by step.

1. **Numerator**:
- \(324 = 18^2\)
- \(640 = 64 \times 10 = 8^2 \times 10\)
- \(129600 = 1296 \times 100 = 36^2 \times 100\)

So, the numerator becomes:

\[ 324 \times 640 \times 129600 = 18^2 \times (8^2 \times 10) \times (36^2 \times 100) \]

2. **Denominator**:
- \(729 = 27^2\)
- \(1024 = 32^2\)
- \(3600 = 60^2\)

So, the denominator becomes:

\[ 729 \times 1024 \times 3600 = 27^2 \times 32^2 \times 60^2 \]

### Step 3: Further simplification
Now we can rewrite the expression as follows:

\[ \frac{(18 \times 8 \times 36)^2 \times 1000}{(27 \times 32 \times 60)^2} \]

### Step 4: Taking the square root
Now we can take the square root of the entire expression:

\[ \sqrt{\frac{(18 \times 8 \times 36)^2 \times 1000}{(27 \times 32 \times 60)^2}} = \frac{18 \times 8 \times 36 \times \sqrt{1000}}{27 \times 32 \times 60} \]

### Step 5: Calculate the values
1. Calculate the numerator:
- \(18 \times 8 = 144\)
- \(144 \times 36 = 5184\)

So, the numerator is:

\[ 5184 \times \sqrt{1000} \]

2. Calculate the denominator:
- \(27 \times 32 = 864\)
- \(864 \times 60 = 51840\)

### Step 6: Final simplification
Now we have:

\[ \frac{5184 \times \sqrt{1000}}{51840} \]

This simplifies to:

\[ \frac{1}{10} \times \sqrt{1000} = \frac{1}{10} \times 10\sqrt{10} = \sqrt{10} \]

### Final Result
Thus, the square root of the original expression is:

\[ \sqrt{10} \]