Question
\(\sqrt{\frac{a}{x}\sqrt{\frac{1}{ax}\frac{\sqrt{a}}{\sqrt{x^{3}}}}}\)ax1axax3the square root of a over x end-fraction the square root of 1 over a x end-fraction the fraction with numerator the square root of a end-root and denominator the square root of x cubed end-root end-fraction end-root end-rootππ₯1ππ₯πβπ₯3ββ·β·
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To simplify the expression
\[
\sqrt{\frac{a}{x}\sqrt{\frac{1}{ax}\frac{\sqrt{a}}{\sqrt{x^{3}}}}}
\]
we'll break it down step by step.
1. **Inside the inner square root**:
Let's first simplify the expression inside the inner square root:
\[
\frac{1}{ax} \cdot \frac{\sqrt{a}}{\sqrt{x^3}} = \frac{\sqrt{a}}{ax\sqrt{x^3}} = \frac{\sqrt{a}}{a\sqrt{x}\cdot x^{3/2}} = \frac{\sqrt{a}}{a x^{2}} = \frac{1}{\sqrt{a}} \cdot \frac{1}{x^{2}}
\]
So the inner square root simplifies to:
\[
\sqrt{\frac{1}{ax}\cdot \frac{\sqrt{a}}{\sqrt{x^3}}} = \sqrt{\frac{1}{a x^2}}
\]
2. **Now substitute back**:
Substitute back into the outer square root:
\[
\sqrt{\frac{a}{x} \cdot \sqrt{\frac{1}{a x^2}}}
\]
This becomes:
\[
\sqrt{\frac{a}{x} \cdot \frac{1}{\sqrt{a x^2}}} = \sqrt{\frac{a}{x} \cdot \frac{1}{\sqrt{a} \sqrt{x^2}}} = \sqrt{\frac{a}{x} \cdot \frac{1}{\sqrt{a}x}} = \sqrt{\frac{1}{\sqrt{a}} \cdot \frac{1}{x^2}}
\]
3. **Combine fractions**:
Combine the components:
\[
\sqrt{\frac{1}{\sqrt{a} \cdot x^2}} = \frac{1}{\sqrt{\sqrt{a} \cdot x^2}} = \frac{1}{\sqrt{a^{1/2} x^2}} = \frac{1}{\sqrt{a^{1/2}} \cdot \sqrt{x^2}} = \frac{1}{\sqrt{a^{1/2}} \cdot x} = \frac{1}{\sqrt{a} x}
\]
So the final simplified form of the original expression is
\[
\frac{1}{\sqrt{a} x}
\]
\[
\sqrt{\frac{a}{x}\sqrt{\frac{1}{ax}\frac{\sqrt{a}}{\sqrt{x^{3}}}}}
\]
we'll break it down step by step.
1. **Inside the inner square root**:
Let's first simplify the expression inside the inner square root:
\[
\frac{1}{ax} \cdot \frac{\sqrt{a}}{\sqrt{x^3}} = \frac{\sqrt{a}}{ax\sqrt{x^3}} = \frac{\sqrt{a}}{a\sqrt{x}\cdot x^{3/2}} = \frac{\sqrt{a}}{a x^{2}} = \frac{1}{\sqrt{a}} \cdot \frac{1}{x^{2}}
\]
So the inner square root simplifies to:
\[
\sqrt{\frac{1}{ax}\cdot \frac{\sqrt{a}}{\sqrt{x^3}}} = \sqrt{\frac{1}{a x^2}}
\]
2. **Now substitute back**:
Substitute back into the outer square root:
\[
\sqrt{\frac{a}{x} \cdot \sqrt{\frac{1}{a x^2}}}
\]
This becomes:
\[
\sqrt{\frac{a}{x} \cdot \frac{1}{\sqrt{a x^2}}} = \sqrt{\frac{a}{x} \cdot \frac{1}{\sqrt{a} \sqrt{x^2}}} = \sqrt{\frac{a}{x} \cdot \frac{1}{\sqrt{a}x}} = \sqrt{\frac{1}{\sqrt{a}} \cdot \frac{1}{x^2}}
\]
3. **Combine fractions**:
Combine the components:
\[
\sqrt{\frac{1}{\sqrt{a} \cdot x^2}} = \frac{1}{\sqrt{\sqrt{a} \cdot x^2}} = \frac{1}{\sqrt{a^{1/2} x^2}} = \frac{1}{\sqrt{a^{1/2}} \cdot \sqrt{x^2}} = \frac{1}{\sqrt{a^{1/2}} \cdot x} = \frac{1}{\sqrt{a} x}
\]
So the final simplified form of the original expression is
\[
\frac{1}{\sqrt{a} x}
\]
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.