Write as a radical expression and evaluate if possible. [tex]$32^{\frac{1}{5}}$[/tex]
A. [tex]$\sqrt[5]{32}=-2$[/tex]
B. [tex]$\sqrt[5]{32}=3$[/tex]
C. [tex]$\sqrt[5]{32}$[/tex], not possible
D. [tex]$\sqrt[5]{32}=2$[/tex]
Answer (2)
Rewrite the expression 3 2 5 1 in radical form as 5 32 .
Evaluate the radical expression: 5 32 = 2 because 2 5 = 32 .
Compare the result with the given options and identify the correct one.
The correct answer is 2 .
Explanation
Rewrite in radical form We are asked to express 3 2 5 1 as a radical expression and evaluate it, if possible. The expression 3 2 5 1 can be rewritten as 5 32 .
Evaluate the radical Now we need to evaluate 5 32 . This means we are looking for a number that, when raised to the power of 5, equals 32. We can express 32 as 2 5 , since 2 × 2 × 2 × 2 × 2 = 32 . Therefore, 5 32 = 5 2 5 = 2 .
Compare with options Comparing our result with the given options: Option A: 5 32 = − 2 is incorrect because ( − 2 ) 5 = − 32 , not 32. Option B: 5 32 = 3 is incorrect because 3 5 = 243 , not 32. Option C: 5 32 , not possible is incorrect because we found that 5 32 = 2 .
Option D: 5 32 = 2 is correct because 2 5 = 32 .
Examples
Understanding radical expressions is crucial in various fields, such as engineering and physics. For instance, when calculating the period of a pendulum, you might encounter a formula involving a square root. Similarly, in electrical engineering, determining the impedance of a circuit can involve complex numbers and radicals. Mastering these concepts provides a solid foundation for tackling more advanced problems in these disciplines. This skill is also useful in simplifying calculations in computer graphics and game development, where efficient computation of distances and transformations is essential.
The expression 3 2 5 1 rewrites to 5 32 and evaluates to 2, since 2 5 = 32 . The correct option is D: 5 32 = 2 .
;
