`root(3)((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 will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sqrt[3]{\frac{72.9}{0.4096}} \]
To simplify, we can multiply the numerator and the denominator by \( 10000 \) (which is \( 10^4 \)) to eliminate the decimal in the numerator and make the denominator a whole number:
\[ \sqrt[3]{\frac{72.9 \times 10000}{0.4096 \times 10000}} = \sqrt[3]{\frac{729000}{4.096}} \]

### Step 2: Convert the denominator to a whole number
Next, we convert \( 4.096 \) to a whole number:
\[ 4.096 = \frac{4096}{1000} \]
Thus, we can rewrite our expression as:
\[ \sqrt[3]{\frac{729000 \times 1000}{4096}} = \sqrt[3]{\frac{729000000}{4096}} \]

### Step 3: Factor the numerator and denominator
Now, we need to factor both the numerator and the denominator:
- The numerator \( 729000000 \) can be factored as:
\[ 729000000 = 729 \times 1000 = 729 \times 10^3 = 9^3 \times 10^3 \]
- The denominator \( 4096 \) can be factored as:
\[ 4096 = 2^{12} \]

### Step 4: Combine the factors
Substituting these factorizations back into our cube root, we have:
\[ \sqrt[3]{\frac{9^3 \times 10^3}{2^{12}}} \]

### Step 5: Simplify the cube root
Now we can simplify the cube root:
\[ \sqrt[3]{9^3} = 9 \quad \text{and} \quad \sqrt[3]{10^3} = 10 \]
Thus,
\[ \sqrt[3]{\frac{9^3 \times 10^3}{2^{12}}} = \frac{9 \times 10}{\sqrt[3]{2^{12}}} = \frac{90}{2^4} = \frac{90}{16} \]

### Step 6: Final simplification
Now we simplify \( \frac{90}{16} \):
\[ \frac{90}{16} = \frac{45}{8} = 5.625 \]

### Conclusion
Thus, the value of \( \sqrt[3]{\frac{72.9}{0.4096}} \) is:
\[ \boxed{5.625} \]