Question

What is the value of the square root of the square root of .00000256?

• A. 0.004
• B. 0.016
• C. 0.04
• D. 0.16
• E. 0.4

Hint:

When taking the square root of a decimal number, count the number of decimal places. If there are \(2n\) places, the root will have \(n\) places. For 0.00000256 (8 decimal places), its square root (0.0016) has 4 decimal places. For 0.0016 (4 decimal places), its square root (0.04) has 2 decimal places.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The question asks to evaluate a nested square root, which means we need to perform the square root operation twice. The expression can be written as \( \sqrt{\sqrt{0.00000256}} \). This is equivalent to finding the fourth root of the number.

Step 2: Key Formula or Approach:
We will solve this by taking the square root two times. It is often easier to handle decimals by converting them to scientific notation or fractions.
\( 0.00000256 = 256 \times 10^{-8} \)

Step 3: Detailed Explanation:
Let's evaluate the expression step-by-step.
First, find the inner square root: \( \sqrt{0.00000256} \).
Using scientific notation: \[ \sqrt{0.00000256} = \sqrt{256 \times 10^{-8}} \] The square root of a product is the product of the square roots: \[ \sqrt{256} \times \sqrt{10^{-8}} \] We know that \( \sqrt{256} = 16 \) and \( \sqrt{10^{-8}} = 10^{-8/2} = 10^{-4} \).
So, the result of the first square root is: \[ 16 \times 10^{-4} = 0.0016 \] Now, we take the square root of this result: \( \sqrt{0.0016} \).
Again, using scientific notation: \[ \sqrt{0.0016} = \sqrt{16 \times 10^{-4}} \] \[ \sqrt{16} \times \sqrt{10^{-4}} \] We know that \( \sqrt{16} = 4 \) and \( \sqrt{10^{-4}} = 10^{-4/2} = 10^{-2} \).
So, the final result is: \[ 4 \times 10^{-2} = 0.04 \] Step 4: Final Answer
The value of the square root of the square root of .00000256 is 0.04.