Express $\sqrt[4]{0.16}$ in standard form

Analysis

To solve this problem, we need to simplify the fourth root of a decimal number and then convert the resulting value into standard form (scientific notation).

Concept

1. Fourth Root: $\sqrt[4]{x}$ is the same as $x^{1/4}$.
2. Decimal to Fraction: Convert $0.16$ to a fraction to make the root easier to calculate: $0.16 = \frac{16}{100} = \frac{4}{25}$ or $\frac{16}{100} = \frac{16}{10^2}$.
3. Standard Form: A number in standard form is written as $a \times 10^n$, where $1 \le a < 10$.

Step-by-Step Solution

Step 1: Express the decimal as a fraction.

So, we are looking for $\sqrt[4]{\frac{16}{100}}$.

Step 2: Apply the root to the numerator and denominator.

Step 3: Simplify the numerator. Since $2^4 = 16$, then $\sqrt[4]{16} = 2$.

Step 4: Simplify the denominator. $\sqrt[4]{100} = (10^2)^{1/4} = 10^{2 \times \frac{1}{4}} = 10^{1/2}$.

Step 5: Combine the results. The expression becomes:

Step 6: Convert to standard form. Using the law of indices, $\frac{1}{a^n} = a^{-n}$:

Conclusion

Comparing our result $2 \times 10^{-1/2}$ to the given options, it matches Option A.

Exam Tip: When dealing with roots of decimals, always convert them to fractions with denominators that are powers of 10 (like 100, 10000). It makes applying the laws of indices much cleaner and prevents errors with decimal point placement!