Sure, let's simplify the given mathematical expression:
[tex]\[ \frac{\sqrt{x^2 - 1} - \frac{1}{2} x (x^2 - 1)^{-1/2}}{x^2 - 1} \][/tex]
To simplify this expression step by step, let's break it down:
1. Write Out the Expression More Clearly:
[tex]\[ \frac{\sqrt{x^2 - 1} - \frac{1}{2} x (x^2 - 1)^{-1/2}}{x^2 - 1} \][/tex]
2. Combine the Terms in the Numerator:
In the numerator, we have two terms: [tex]\(\sqrt{x^2 - 1}\)[/tex] and [tex]\(- \frac{1}{2} x (x^2 - 1)^{-1/2}\)[/tex]. Let's write the numerator in a single fraction:
[tex]\[ \sqrt{x^2 - 1} - \frac{1}{2} x (x^2 - 1)^{-1/2} \][/tex]
The numerator is already in its simplest form, so let’s proceed to simplify the entire expression by combining the terms.
3. Rewrite the Numerator:
Notice that [tex]\(\sqrt{x^2 - 1}\)[/tex] is [tex]\( (x^2 - 1)^{1/2} \)[/tex] and [tex]\((x^2 - 1)^{-1/2} \)[/tex] can be written as [tex]\(\frac{1}{(x^2 - 1)^{1/2}}\)[/tex]. So we have:
[tex]\[ (x^2 - 1)^{1/2} - \frac{1}{2} x (x^2 - 1)^{-1/2} \][/tex]
4. Factor Out Common Terms (if possible):
There isn't a straightforward way to factor out common terms in the numerator as they are not directly factorizable.
5. Simplify the Whole Fraction:
We need to simplify the fraction:
[tex]\[ \frac{(x^2 - 1)^{1/2} - 0.5x (x^2 - 1)^{-1/2}}{x^2 - 1} \][/tex]
Notice the term in the denominator [tex]\( x^2 - 1 \)[/tex] can be written as [tex]\( (x^2 - 1) \)[/tex].
6. Rewrite Using Simplified Components:
So the fraction can be expressed as:
[tex]\[ \frac{\sqrt{x^2 - 1} - \frac{1}{2} x \frac{1}{\sqrt{x^2 - 1}}}{x^2 - 1} \][/tex]
Let's rewrite it step by step:
[tex]\[ \text{Numerator: } \sqrt{x^2 - 1} - \frac{1}{2} x (x^2 - 1)^{-1/2} \][/tex]
[tex]\[ = \sqrt{x^2 - 1} - \frac{1}{2} x \cdot \frac{1}{\sqrt{x^2 - 1}} \][/tex]
[tex]\[ = \sqrt{x^2 - 1} - \frac{1}{2} \cdot \frac{x}{\sqrt{x^2 - 1}} \][/tex]
[tex]\[ \text{Combine under a common denominator: } \frac{\sqrt{x^2 - 1} \cdot \sqrt{x^2 - 1} - \frac{1}{2} x}{\sqrt{x^2 - 1}} \][/tex]
7. Express the Entire Fraction Clearly:
Combine the terms together:
[tex]\[ \frac{(x^2 - 1) - 0.5x}{\sqrt{x^2 - 1} \cdot (x^2 - 1)} \][/tex]
Therefore, the final simplified form would be written in terms of the expressions:
[tex]\[ \boxed{\frac{-0.5x + (x^2 - 1)}{(x^2 - 1)^{1.5}}} \][/tex]
So, the given expression simplifies to:
[tex]\[ \boxed{\frac{-0.5x + (x^2 - 1)^{1.0}}{(x^2 - 1)^{1.5}}} \][/tex]
