Which expression is equivalent to $\frac{\sqrt{10}}{\sqrt[4]{8}}$ ?
A. $\frac{\sqrt[4]{200}}{2}$
B. $\frac{\sqrt[4]{20}}{2}$
C. $\frac{2 \sqrt{5}}{5}$
D. $\frac{100}{8}
Answer (2)
The expression 4 8 10 simplifies to 2 4 200 , which matches option A. Therefore, the equivalent expression is A ) 2 4 200 .
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Rewrite the original expression using exponents.
Simplify the expression by breaking down the terms and rationalizing the denominator.
Combine the terms under a single radical.
The simplified expression is 2 4 200 .
Explanation
Understanding the Problem We are given the expression 4 8 10 and four possible equivalent expressions. Our goal is to determine which of the four expressions is equivalent to the given expression.
Rewriting with Exponents Let's rewrite the given expression using exponents: 4 8 10 = 8 1/4 1 0 1/2 .
Simplifying the Denominator Now, rewrite 8 as 2 3 , so the expression becomes ( 2 3 ) 1/4 1 0 1/2 = 2 3/4 1 0 1/2 .
Breaking Down the Numerator Rewrite 10 as 2 ⋅ 5 , so the expression becomes 2 3/4 ( 2 ⋅ 5 ) 1/2 = 2 3/4 2 1/2 ⋅ 5 1/2 = 2 1/2 − 3/4 ⋅ 5 1/2 = 2 − 1/4 ⋅ 5 1/2 = 2 1/4 5 1/2 .
Rationalizing the Denominator To rationalize the denominator, multiply the numerator and denominator by 2 3/4 : 2 1/4 ⋅ 2 3/4 5 1/2 ⋅ 2 3/4 = 2 5 1/2 ⋅ 2 3/4 = 2 5 ⋅ 4 8 = 2 5 ⋅ 4 2 3 .
Combining Terms Rewrite the numerator as 5 ⋅ 4 2 3 = 4 5 2 ⋅ 4 2 3 = 4 5 2 ⋅ 2 3 = 4 25 ⋅ 8 = 4 200 .
Final Simplification Therefore, the expression simplifies to 2 4 200 . Comparing this result to the four given options, we find that it matches the first option.
Conclusion The expression 4 8 10 is equivalent to 2 4 200 .
Examples
Understanding how to simplify radical expressions is useful in various fields, such as engineering and physics, where complex calculations involving roots and exponents are common. For example, when calculating the impedance of an electrical circuit or determining the energy levels of quantum particles, simplifying radical expressions can make the calculations more manageable and lead to more accurate results. This skill also helps in fields like computer graphics, where efficient calculations involving square roots are essential for rendering images and creating realistic visual effects. Simplifying radicals allows for more efficient computation and better performance in these applications.
