Certainly! Let's solve the equation [tex]\(2 x^{-\frac{1}{2}} - 2 = 0\)[/tex] step by step.
1. Isolate the term involving [tex]\(x\)[/tex]:
The given equation is:
[tex]\[
2 x^{-\frac{1}{2}} - 2 = 0
\][/tex]
Add 2 to both sides of the equation to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[
2 x^{-\frac{1}{2}} = 2
\][/tex]
2. Divide both sides by 2:
Next, divide both sides of the equation by 2 to further simplify:
[tex]\[
x^{-\frac{1}{2}} = 1
\][/tex]
3. Rewrite the exponent:
Recall that [tex]\(x^{-\frac{1}{2}}\)[/tex] is the same as [tex]\(1 / \sqrt{x}\)[/tex]. To eliminate the negative exponent, we rewrite the equation:
[tex]\[
x^{-\frac{1}{2}} = (\sqrt{x})^{-1} = 1
\][/tex]
4. Reciprocal properties:
Since [tex]\((x^{a})^{-1} = 1 / x^a\)[/tex], we know that [tex]\(\sqrt{x}\)[/tex] must be such that its reciprocal equals 1:
[tex]\[
\frac{1}{\sqrt{x}} = 1
\][/tex]
5. Solve for [tex]\(\sqrt{x}\)[/tex]:
Take the reciprocal of both sides to get:
[tex]\[
\sqrt{x} = 1
\][/tex]
6. Square both sides:
To solve for [tex]\(x\)[/tex], square both sides of the equation:
[tex]\[
(\sqrt{x})^2 = 1^2
\][/tex]
Simplifying, we have:
[tex]\[
x = 1
\][/tex]
The solution to the equation [tex]\(2 x^{-\frac{1}{2}} - 2 = 0\)[/tex] is:
[tex]\[
x = 1
\][/tex]
