Q. If \(x = \sqrt{\frac{25}{16}}\). What is the value of \(\sqrt{\frac{5}{x}}\)?
Answer
\(x=\sqrt{\frac{25}{16}}\)
Take the square root of the numerator and the denominator separately.
\(x=\frac{\sqrt{25}}{\sqrt{16}}\)
\(\sqrt{25}=5\) and \(\sqrt{16}=4\), so:
\(x=\frac{5}{4}\)
Now simplify the expression.
\(\sqrt{\frac{5}{x}}\)
Substitute \(x=\frac{5}{4}\).
\(\sqrt{\frac{5}{\frac{5}{4}}}\)
Dividing by a fraction means multiplying by its reciprocal.
\(\sqrt{5\cdot\frac{4}{5}}\)
Simplify inside the square root.
\(\sqrt{4}\)
\(\sqrt{4}=2\)
Final result: \(2\)
Detailed Explanation
- Start with the given expression for x:\(x = \sqrt{\dfrac{25}{16}}\)
- Use the property \(\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}\) to simplify:\(x = \dfrac{\sqrt{25}}{\sqrt{16}}\)
- Evaluate the square roots (principal, nonnegative roots):\(x = \dfrac{5}{4}\)
- Form the required expression and substitute the value of \(x\):\(\sqrt{\dfrac{5}{x}} = \sqrt{\dfrac{5}{5/4}}\)
- Simplify the fraction inside the square root by multiplying by the reciprocal:\(\dfrac{5}{5/4} = 5 \cdot \dfrac{4}{5} = 4\)
- Take the square root of 4:\(\sqrt{4} = 2\)
- Conclusion:\(\sqrt{\dfrac{5}{x}} = 2\)
See full solution
Algebra FAQs
What is x if \( x = \sqrt{\frac{25}{16}} \)?.
\(x = \sqrt{25/16} = \sqrt{(5/4)^2} = 5/4\), because the principal square root is nonnegative.
What is the value of \(\sqrt{5/x}\) given \(x = \sqrt{25/16}\)?
Substitute \(x = \frac{5}{4}\): \(\sqrt{\frac{5}{(5/4)}} = \sqrt{4} = 2\).
Could \(x\) be \(-5/4\) instead of \(5/4\)?
No. By convention \( \sqrt{\,\cdot\,} \) denotes the principal (nonnegative) root, so \( \sqrt{25/16}=5/4 \), not \(-5/4\)..
Why is \( \sqrt{a^2} = |a| \) and not \( a \)?
Because the square root function returns the nonnegative value whose square is a^2, so \( \sqrt{a^2} = |a| \), ensuring a nonnegative result.
Are there domain restrictions for \( \sqrt{5/x} \)?
Yes. The radicand \(5/x\) must be \(\ge 0\) and \(x \ne 0\). For real numbers this requires \(x>0\) (since \(5>0\)), so \(x=5/4\) is valid.
How to simplify using exponents instead of radicals?
How to simplify using exponents instead of radicals?
How to check your result is correct?
Recompute: \(x=5/4\), \(5/x=5/(5/4)=4\), \(\sqrt{4}=2\). Optionally square your answer: \(2^2=4\) equals \(5/x\), so result checks.
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