Of course! Let's go through the steps to simplify and understand the given expression [tex]\(\left(x^2 + x + \frac{1}{2}\right)^{-1/2}\)[/tex] in detail.
### Step-by-Step Solution:
1. Identify the Expression:
The given mathematical expression is [tex]\(\left(x^2 + x + \frac{1}{2}\right)^{-1/2}\)[/tex].
2. Exponentiation Rule:
The expression [tex]\(\left(x^2 + x + \frac{1}{2}\right)^{-1/2}\)[/tex] involves an exponent of [tex]\(-1/2\)[/tex]. According to exponentiation rules, an expression [tex]\(a^{-n}\)[/tex] is equivalent to [tex]\(\frac{1}{a^{n}}\)[/tex]. Therefore, we can rewrite the given expression using this rule.
3. Rewrite the Expression:
Applying the exponentiation rule, the expression [tex]\(\left(x^2 + x + \frac{1}{2}\right)^{-1/2}\)[/tex] can be rewritten as:
[tex]\[
\left(x^2 + x + \frac{1}{2}\right)^{-1/2} = \frac{1}{\left(x^2 + x + \frac{1}{2}\right)^{1/2}}
\][/tex]
4. Simplification:
The term [tex]\(\left(x^2 + x + \frac{1}{2}\right)^{1/2}\)[/tex] denotes the square root of [tex]\(x^2 + x + \frac{1}{2}\)[/tex]. Therefore, our expression can now be written as:
[tex]\[
\frac{1}{\sqrt{x^2 + x + \frac{1}{2}}}
\][/tex]
5. Combine Steps:
Combining the rewriting and simplification steps, we get the final form:
[tex]\[
\left(x^2 + x + \frac{1}{2}\right)^{-1/2} = \frac{1}{\sqrt{x^2 + x + \frac{1}{2}}}
\][/tex]
### Conclusion:
The simplified form of the expression [tex]\(\left(x^2 + x + \frac{1}{2}\right)^{-1/2}\)[/tex] is:
[tex]\[
\frac{1}{\sqrt{x^2 + x + \frac{1}{2}}}
\][/tex]
So, the original expression [tex]\(\left(x^2+x+\frac{1}{2}\right)^{-1 / 2}\)[/tex] evaluates to [tex]\(\frac{1}{\sqrt{x^2 + x + \frac{1}{2}}}\)[/tex].
