Simplify the following expression: $\frac{\sqrt{3}}{\sqrt{24}}$
Understand the Problem
The question asks to simplify the given expression, which involves a fraction with square roots in the numerator and denominator. The goal is to simplify the expression by rationalizing the denominator and reducing the expression to its simplest form.
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: $24 = 2 \cdot 2 \cdot 2 \cdot 3 = 2^3 \cdot 3$. So, $\sqrt{24} = \sqrt{2^2 \cdot 2 \cdot 3} = 2\sqrt{6}$. Therefore, the original expression becomes $\frac{\sqrt{3}}{2\sqrt{6}}$.
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Rationalize the denominator To rationalize the denominator, multiply both the numerator and denominator by $\sqrt{6}$: $$ \frac{\sqrt{3}}{2\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} = \frac{\sqrt{3 \cdot 6}}{2 \cdot 6} = \frac{\sqrt{18}}{12} $$
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Simplify the numerator Simplify $\sqrt{18}$: $18 = 2 \cdot 3 \cdot 3 = 2 \cdot 3^2$. So, $\sqrt{18} = \sqrt{3^2 \cdot 2} = 3\sqrt{2}$. Therefore, the expression becomes $\frac{3\sqrt{2}}{12}$.
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Reduce the fraction Divide both the numerator and denominator by their greatest common divisor, which is 3: $$ \frac{3\sqrt{2}}{12} = \frac{3\sqrt{2} \div 3}{12 \div 3} = \frac{\sqrt{2}}{4} $$
$\frac{\sqrt{2}}{4}$
More Information
The simplified form of $\frac{\sqrt{3}}{\sqrt{24}}$ is $\frac{\sqrt{2}}{4}$. Rationalizing the denominator is a standard technique used to simplify expressions with radicals in the denominator, making them easier to work with.
Tips
A common mistake is not simplifying the square roots completely before or after rationalizing the denominator. Another mistake is incorrectly multiplying or dividing within the square roots. For example, some might forget to multiply both the number outside the square root and the number inside when rationalizing.
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