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

Answer Details

To express \(\sqrt[4]{0.16}\) in standard form, let's break the problem into clear steps:

• Step 1: Rewrite the decimal in standard form.
  0.16 can be written as \(1.6 \times 10^{-1}\). This is because standard form has the structure \(a \times 10^n\) where \(1 \leq a < 10\).
• Step 2: Apply the fourth root to the expression.
  \[ \sqrt[4]{0.16} = \sqrt[4]{1.6 \times 10^{-1}} \] By root laws: \[ \sqrt[4]{ab} = \sqrt[4]{a} \times \sqrt[4]{b} \] So: \[ \sqrt[4]{1.6 \times 10^{-1}} = \sqrt[4]{1.6} \times \sqrt[4]{10^{-1}} \]
• Step 3: Simplify each part separately.
  
  • For \(\sqrt[4]{10^{-1}}\):
    The property \(\sqrt[n]{x^m} = x^{m/n}\) gives: \[ \sqrt[4]{10^{-1}} = 10^{-\frac{1}{4}} \]
  • For \(\sqrt[4]{1.6}\):
    1.6 is not a perfect fourth power, so let's express 0.16 as a fraction instead. Notice that: \[ 0.16 = \frac{16}{100} \]
• Step 4: Find the fourth root of the fraction:
  \[ \sqrt[4]{\frac{16}{100}} = \frac{\sqrt[4]{16}}{\sqrt[4]{100}} \] \[ \sqrt[4]{16} = 2 \quad \text{(since } 2^4 = 16 \text{)} \] \[ \sqrt[4]{100} = (100)^{1/4} = (10^2)^{1/4} = 10^{\frac{2}{4}} = 10^{\frac{1}{2}} \] So: \[ \sqrt[4]{0.16} = \frac{2}{10^{1/2}} \]
• Step 5: Express in standard form.
  Divide by \(10^{1/2}\) (which is \(\sqrt{10}\)), or rewrite the fraction in index form: \[ \frac{2}{10^{1/2}} = 2 \times 10^{-1/2} \] This matches the standard form.

Summary:
The correct answer is \(2 \times 10^{-\frac{1}{2}}\).
This is because the fourth root of 0.16 is the same as dividing 2 by the square root of 10, which is most compactly expressed using indices as \(2 \times 10^{-1/2}\).