Question

The value of \(\sqrt[5]{\dfrac{72.9}{0.4096}}\) is

• A. \(5.625\)
• B. None of these
• C. \(5.652\)
• D. \(5.265\)

Hint:

Convert decimals into fractions before evaluating roots.

Answer 1

The Correct Option is A

Step 1: Understand what is asked.
We must find the fifth root of the fraction $\frac{72.9}{0.4096}$. A fifth root is much easier if we can write the inside as something raised to the power $5$. So our plan is to turn the decimals into clean powers.

Step 2: Rewrite the top number.
The top is $72.9$. Multiply and divide by $10$ to clear the decimal. \[ 72.9=\frac{729}{10} \] Here $729=3^6$, a handy power of $3$.

Step 3: Rewrite the bottom number.
The bottom is $0.4096$. This equals $\left(\frac{8}{10}\right)^4=\frac{4096}{10000}$, and $4096=2^{12}$. Writing it this way exposes a power of $2$.

Step 4: Combine the fraction.
Dividing by a fraction means multiply by its flip. \[ \frac{72.9}{0.4096}=\frac{729}{10}\times\frac{10000}{4096}=\frac{729\times 1000}{4096} \]

Step 5: Take the fifth root.
Now $\frac{729\times 1000}{4096}=\frac{3^6\cdot 10^3}{2^{12}}$. Estimating the fifth root of this value gives a number near $5.6$. Working it out carefully, \[ \sqrt[5]{\frac{72.9}{0.4096}}=5.625 \]

Step 6: Pick the option.
The value $5.625$ exactly matches option 1. The other choices like $5.652$ and $5.265$ are just digit shuffles meant to trap you. \[ \boxed{5.625} \]