Simplify $\frac{\sqrt{3}}{\sqrt{24}}$
Understand the Problem
The question requires simplifying the given expression which involves radicals. To simplify, we need to rationalize the denominator and potentially simplify the radicals further.
Answer
$\frac{\sqrt{2}}{4}$
Answer for screen readers
$\frac{\sqrt{2}}{4}$
Steps to Solve
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Simplify the denominator Simplify $\sqrt{24}$ by finding its prime factors: $24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3$. So, $\sqrt{24} = \sqrt{2^2 \times 2 \times 3} = 2\sqrt{6}$. The expression becomes $\frac{\sqrt{3}}{2\sqrt{6}}$.
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Rationalize the denominator Multiply both the numerator and the denominator by $\sqrt{6}$:
$\frac{\sqrt{3}}{2\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{\sqrt{3} \times \sqrt{6}}{2 \times 6}$
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Simplify the numerator Simplify $\sqrt{3} \times \sqrt{6}$: $\sqrt{3} \times \sqrt{6} = \sqrt{18} = \sqrt{2 \times 3^2} = 3\sqrt{2}$. The expression becomes $\frac{3\sqrt{2}}{12}$.
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Reduce the fraction Simplify the fraction $\frac{3\sqrt{2}}{12}$ by dividing both the numerator and denominator by 3 which gives us $\frac{\sqrt{2}}{4}$.
$\frac{\sqrt{2}}{4}$
More Information
The simplified form of $\frac{\sqrt{3}}{\sqrt{24}}$ is $\frac{\sqrt{2}}{4}$.
Tips
A common mistake is not simplifying the radicals completely or making errors while rationalizing the denominator. For example, incorrectly simplifying $\sqrt{24}$ or not multiplying both the numerator and the denominator by the same value when rationalizing.
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