Given x, is greater than, 0x>0 and y, is greater than, 0, commay>0, select the expression that is equivalent to

Question

square root of, minus, 100, x, y, to the power 8 , end square root
−100xy
8

​

Answer
Multiple Choice Answers
10, i, x, squared, y, start superscript, one quarter, end superscript10ix
2
y
4
1
​


minus, 10, x, squared, y, start superscript, one quarter, end superscript−10x
2
y
4
1
​


minus, 10, x, start superscript, one half, end superscript, y, to the power 4−10x
2
1
​

y
4

10, i, x, start superscript, one half, end superscript, y, to the power 410ix
2
1
​

y
4

Answer 1

To simplify the expression \(\sqrt{-100xy^8}\), we start by breaking it down:

1. First, recognize the negative sign within the square root: \[ \sqrt{-100xy^8} = \sqrt{100} \sqrt{-1} \sqrt{xy^8} \]

2. We know that \(\sqrt{100} = 10\) and \(\sqrt{-1} = i\), so: \[ \sqrt{-100xy^8} = 10i \sqrt{xy^8} \]

3. Now simplify \(\sqrt{xy^8}\):

   • \(\sqrt{x}\) is \(x^{1/2}\)
   • \(\sqrt{y^8}\) is \(y^{8/2} = y^4\) Thus: \[ \sqrt{xy^8} = \sqrt{x} \cdot \sqrt{y^8} = x^{1/2} y^4 \]

4. Putting this all together, we have: \[ \sqrt{-100xy^8} = 10i \cdot x^{1/2} y^4 \]

This simplifies to: \[ 10ix^{1/2}y^4 \]

Now, comparing this expression with the provided multiple-choice answers:

1. \(10i \cdot x^2 \cdot y^{1/4}\)
2. \(-10 \cdot x^2 \cdot y^{1/4}\)
3. \(-10 \cdot x^{1/2} \cdot y^4\)
4. \(10i \cdot x^{1/2} \cdot y^4\)

The correct choice is 4) \(10i \cdot x^{1/2} \cdot y^4\).