Alright, let's simplify the expression [tex]\(\frac{2 a}{\sqrt{2 a x}}\)[/tex] step by step.
1. Start with the original expression:
[tex]\[
\frac{2 a}{\sqrt{2 a x}}
\][/tex]
2. Observe that the terms inside the square root can be separated into their individual square roots:
[tex]\[
\sqrt{2 a x} = \sqrt{2} \cdot \sqrt{a} \cdot \sqrt{x}
\][/tex]
Thus, the expression becomes:
[tex]\[
\frac{2 a}{\sqrt{2} \cdot \sqrt{a} \cdot \sqrt{x}}
\][/tex]
3. Separate the terms in the numerator and the denominator:
[tex]\[
\frac{2 a}{\sqrt{2} \cdot \sqrt{a} \cdot \sqrt{x}} = \frac{2 a}{\sqrt{2} \sqrt{a} \sqrt{x}} = \frac{2 a}{\sqrt{2} \cdot \sqrt{a} \cdot \sqrt{x}}
\][/tex]
4. Simplify the fraction by dividing [tex]\(2a\)[/tex] by [tex]\(\sqrt{a}\)[/tex]:
[tex]\[
\frac{2 a}{\sqrt{2} \cdot \sqrt{a} \cdot \sqrt{x}} = \frac{2 \cdot a}{\sqrt{a}} \cdot \frac{1}{\sqrt{2} \sqrt{x}}
\][/tex]
5. Simplify [tex]\(\frac{a}{\sqrt{a}}\)[/tex]:
[tex]\[
\frac{a}{\sqrt{a}} = \frac{a \cdot a^{-1/2}}{1} = \frac{a^{1}}{a^{1/2}} = a^{1 - 1/2} = a^{1/2}
\][/tex]
So, the expression becomes:
[tex]\[
\frac{2 \cdot a^{1/2}}{\sqrt{2} \cdot \sqrt{x}} = \frac{2 \cdot \sqrt{a}}{\sqrt{2} \cdot \sqrt{x}}
\][/tex]
6. Simplify [tex]\(\frac{2}{\sqrt{2}}\)[/tex]:
[tex]\[
\frac{2}{\sqrt{2}} = \frac{2}{2^{1/2}} = 2^{1 - 1/2} = 2^{1/2} = \sqrt{2}
\][/tex]
7. Combine all the simplified terms:
[tex]\[
\frac{\sqrt{2} \cdot \sqrt{a}}{\sqrt{x}} = \frac{\sqrt{2a}}{\sqrt{x}}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\frac{\sqrt{2} \cdot \sqrt{a}}{\sqrt{x}}
\][/tex]
