Change the following to radical form: [tex]x^{\frac{3}{5}}[/tex]
A) [tex]\sqrt{x^{\frac{3}{5}}}[/tex]
B) [tex]x \sqrt{\frac{2}{5}}[/tex]
C) [tex]\sqrt[3]{x^6}[/tex]
D) [tex]\sqrt[5]{x^3}[/tex]
Answer (2)
Convert the fractional exponent to radical form using the rule: x b a = b x a .
Identify a = 3 and b = 5 in the expression x 5 3 .
Apply the rule to get x 5 3 = 5 x 3 .
The correct answer is 5 x 3 .
Explanation
Understanding the problem We are asked to convert the expression x 5 3 into radical form.
Fractional exponents and radicals Recall that a fractional exponent can be expressed as a radical. The general rule is: x b a = b x a where a is the power and b is the root.
Applying the rule In our case, we have x 5 3 . Comparing this to the general form x b a , we can identify a = 3 and b = 5 . Therefore, we can rewrite the expression as: x 5 3 = 5 x 3
Finding the correct option Now, we compare our result with the given options: A) x 5 3 B) x 5 2 C) 3 x 6 D) 5 x 3 Our result, 5 x 3 , matches option D.
Final Answer Therefore, the correct answer is D.
Examples
Understanding radical forms is essential in various fields, such as physics and engineering, where you might encounter expressions involving roots and exponents. For example, when calculating the period of a pendulum, you might use the formula T = 2 π g L , where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity. Converting between radical and exponential forms allows for easier manipulation and simplification of such formulas, making calculations more straightforward. This skill is also crucial in advanced mathematics, such as calculus and differential equations, where manipulating expressions with radicals and exponents is a common task.
To convert x 5 3 to radical form, we apply the rule x b a = b x a , resulting in x 5 3 = 5 x 3 . The correct answer from the options given is D) 5 x 3 .
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