Write $\sqrt[5]{c^8}$ as an expression with a rational exponent.
A. $c^{\frac{5}{8}}$
B. $c^{40}$
C. $c^{\frac{8}{5}}$
D. $c^{13}$
Answer (1)
Recognize the expression as a radical.
Recall the property n x m = x n m .
Apply the property to rewrite the expression: 5 c 8 = c 5 8 .
The expression with a rational exponent is c 5 8 .
Explanation
Understanding the Problem We are given the expression 5 c 8 and asked to rewrite it as an expression with a rational exponent.
Recalling the Property Recall the property that n x m = x n m . This property allows us to convert between radical expressions and expressions with rational exponents.
Applying the Property Applying this property to the given expression 5 c 8 , we identify n = 5 and m = 8 . Therefore, we can rewrite the expression as c 5 8 .
Final Answer Thus, 5 c 8 = c 5 8 .
Examples
Rational exponents are useful in various fields, such as physics and engineering, for simplifying complex equations and calculations. For example, when dealing with wave equations or calculating stress and strain in materials, rational exponents can help in expressing relationships between variables in a more manageable form. Understanding how to convert between radicals and rational exponents allows for easier manipulation and simplification of these equations, leading to more efficient problem-solving.
