Answer :
To express the function \( h(x) \), we start with the given expression:
[tex]\[ h(x) = -\frac{5}{(\sqrt{x} - x)^3} \][/tex]
Let's break this down step-by-step to understand and confirm the function.
1. Identify the components within the fraction:
- The numerator is \( -5 \).
- The denominator is \( (\sqrt{x} - x)^3 \).
2. Simplify the inner expression (denominator):
- The term \( \sqrt{x} \) represents the square root of \( x \).
- The term \( x \) represents the variable itself.
- We are subtracting \( x \) from \( \sqrt{x} \), which results in \( \sqrt{x} - x \).
3. Deal with the exponent:
- Once we have \( \sqrt{x} - x \), we need to raise this entire expression to the power of 3.
- Expressed mathematically, this is \((\sqrt{x} - x)^3\).
4. Combine the components:
- Now that we have broken down the components, we place the numerator \(-5\) over the denominator \((\sqrt{x} - x)^3\).
Putting it all together, we get:
[tex]\[ h(x) = -\frac{5}{(\sqrt{x} - x)^3} \][/tex]
This is the complete and simplified form of the given function [tex]\( h(x) \)[/tex].
[tex]\[ h(x) = -\frac{5}{(\sqrt{x} - x)^3} \][/tex]
Let's break this down step-by-step to understand and confirm the function.
1. Identify the components within the fraction:
- The numerator is \( -5 \).
- The denominator is \( (\sqrt{x} - x)^3 \).
2. Simplify the inner expression (denominator):
- The term \( \sqrt{x} \) represents the square root of \( x \).
- The term \( x \) represents the variable itself.
- We are subtracting \( x \) from \( \sqrt{x} \), which results in \( \sqrt{x} - x \).
3. Deal with the exponent:
- Once we have \( \sqrt{x} - x \), we need to raise this entire expression to the power of 3.
- Expressed mathematically, this is \((\sqrt{x} - x)^3\).
4. Combine the components:
- Now that we have broken down the components, we place the numerator \(-5\) over the denominator \((\sqrt{x} - x)^3\).
Putting it all together, we get:
[tex]\[ h(x) = -\frac{5}{(\sqrt{x} - x)^3} \][/tex]
This is the complete and simplified form of the given function [tex]\( h(x) \)[/tex].
