Which expression is equivalent to \( \frac{\sqrt{10}}{4\sqrt{8}} \)?
Understand the Problem
The question is asking which of the given expressions is equivalent to the fraction involving square roots, specifically ( \frac{\sqrt{10}}{4\sqrt{8}} ). This requires simplification of the expressions to find equality.
Answer
The simplified form of \( \frac{\sqrt{10}}{4\sqrt{8}} \) is \( \frac{\sqrt{5}}{8} \).
Answer for screen readers
The equivalent expression is ( \frac{\sqrt{5}}{8} ).
Steps to Solve
- Simplify the Original Expression
Start with the expression ( \frac{\sqrt{10}}{4\sqrt{8}} ).
- Simplify the Denominator
Rewrite ( \sqrt{8} ) as ( \sqrt{4 \times 2} = 2\sqrt{2} ).
Thus, the denominator becomes: $$ 4\sqrt{8} = 4 \times 2\sqrt{2} = 8\sqrt{2} $$
- Rewrite the Full Expression
Now, the expression looks like this: $$ \frac{\sqrt{10}}{8\sqrt{2}} $$
- Rationalize the Denominator
To rationalize, multiply the numerator and the denominator by ( \sqrt{2} ): $$ \frac{\sqrt{10} \cdot \sqrt{2}}{8\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{20}}{16} $$
- Simplify ( \sqrt{20} )
Rewrite ( \sqrt{20} ) as ( \sqrt{4 \times 5} = 2\sqrt{5} ): $$ \frac{2\sqrt{5}}{16} $$
- Simplify the Fraction
Now, reduce ( \frac{2\sqrt{5}}{16} ) to: $$ \frac{\sqrt{5}}{8} $$
The equivalent expression is ( \frac{\sqrt{5}}{8} ).
More Information
The simplified form of the given expression shows how radicals can be manipulated and simplified, particularly when dealing with square roots in the numerator and denominator.
Tips
- Forgetting to simplify the radicals properly can lead to incorrect conclusions.
- Not rationalizing the denominator when necessary can leave the answer in a less simplified form.
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