Given x, is greater than, 0x>0 and y, is greater than, 0, commay>0, select the expression that is equivalent to
square root of, minus, 100, x, to the power 4 , y, to the power 13 , end square root
−100x 
4
 y 
13
 
​	
 

Given that \( x > 0 \) and \( y > 0 \), we are asked to find an expression equivalent to:

\[
\sqrt{-100x^4y^{13}}
\]

First, let's break down the expression inside the square root:

1. **Factor the expression inside the square root:**

   \[
   -100x^4y^{13} = -1 \times 100 \times x^4 \times y^{13}
   \]

2. **Take the square root of each component:**

   \[
   \sqrt{-100x^4y^{13}} = \sqrt{-1 \times 100 \times x^4 \times y^{13}}
   \]

3. **Simplify each part:**
   - The square root of \(-1\) is \( i \), the imaginary unit.
   - The square root of \( 100 \) is \( 10 \).
   - The square root of \( x^4 \) is \( x^2 \) because \( \sqrt{x^4} = (x^2)^2 = x^2 \).
   - The square root of \( y^{13} \) is \( y^{6.5} \), which can be rewritten as \( y^6 \sqrt{y} \).

So, the expression simplifies to:

\[
\sqrt{-100x^4y^{13}} = 10x^2y^6\sqrt{y} \times i
\]

Thus, the expression equivalent to the given expression is:

\[
10x^2y^6\sqrt{y} \times i
\]

This is the simplified form of the expression.

Would you like further details or have any questions? 

Here are 5 related questions to expand on this concept:

1. How would you handle a similar problem if \( x \) or \( y \) were negative?
2. What is the significance of the imaginary unit \( i \) in complex numbers?
3. Can you simplify the square root of a product involving more variables?
4. How do you determine the square root of a higher power of a variable, like \( y^{13} \)?
5. What are some real-world applications where complex numbers are used?

**Tip:** When dealing with square roots of negative numbers, always consider the imaginary unit \( i \), where \( i = \sqrt{-1} \).

Explore the concept of square roots and imaginary numbers in mathematics with a focus on solving complex expressions involving exponents and radicals. This page provides a detailed explanation of how to simplify expressions like \( \sqrt{-100x^4y^{13}} \) and their applications. Suitable for Grades 10-12.

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