`root3(72.9/0.4096)` is equal to :

Options

A. 0.5625
B. 5.625
C. 182
D. 13.6

Answer

To solve the problem \( \sqrt[3]{\frac{72.9}{0.4096}} \), we can follow these steps:

### Step 1: Simplify the fraction
We start by rewriting the expression:
\[ \frac{72.9}{0.4096} \]
To simplify this, we can multiply both the numerator and the denominator by 10000 to eliminate the decimal in the denominator:
\[ \frac{72.9 \times 10000}{0.4096 \times 10000} = \frac{729000}{4096} \]

### Step 2: Factor the numbers
Next, we can factor both the numerator and the denominator:
- \( 729 = 9^3 = (3^2)^3 = 3^6 \)
- \( 4096 = 16^3 = (2^4)^3 = 2^{12} \)

So we can rewrite the fraction as:
\[ \frac{729000}{4096} = \frac{729 \times 1000}{4096} = \frac{729 \times 10^3}{4096} \]

### Step 3: Rewrite \( 1000 \) and \( 4096 \)
Now, we can express \( 1000 \) and \( 4096 \) in terms of their prime factors:
- \( 1000 = 10^3 = (2 \times 5)^3 = 2^3 \times 5^3 \)
- \( 4096 = 2^{12} \)

Thus, we can express the fraction as:
\[ \frac{729 \times (2^3 \times 5^3)}{2^{12}} = \frac{729 \times 2^3 \times 5^3}{2^{12}} = 729 \times 5^3 \times \frac{1}{2^9} \]

### Step 4: Calculate the cube root
Now we can take the cube root of the entire expression:
\[ \sqrt[3]{729 \times 5^3 \times \frac{1}{2^9}} = \sqrt[3]{729} \times \sqrt[3]{5^3} \times \sqrt[3]{\frac{1}{2^9}} \]
Calculating each part:
- \( \sqrt[3]{729} = 9 \)
- \( \sqrt[3]{5^3} = 5 \)
- \( \sqrt[3]{\frac{1}{2^9}} = \frac{1}{2^3} = \frac{1}{8} \)

Putting it all together:
\[ 9 \times 5 \times \frac{1}{8} = \frac{45}{8} \]

### Final Answer
Thus, the value of \( \sqrt[3]{\frac{72.9}{0.4096}} \) is:
\[ \frac{45}{8} \]