Select the correct answer. Which expression is equivalent to [tex]$\sqrt[6]{g^5}$[/tex], if [tex]$g\ \textgreater \ 0$[/tex]?
A. [tex]$g^{\frac{6}{3}}$[/tex]
B. [tex]$5 g^6$[/tex]
C. [tex]$\frac{5}{6} g$[/tex]
D. [tex]$g^{\frac{5}{6}}$[/tex]
Answer (1)
The problem requires converting a radical expression to an equivalent expression with a fractional exponent. Using the property n x m = x n m , we rewrite 6 g 5 as g 6 5 . The correct answer is g 6 5 .
Explanation
Understanding the problem We are given the expression 6 g 5 and asked to find an equivalent expression from the given options, knowing that 0"> g > 0 .
Recalling the properties of radicals and exponents Recall the property of radicals and exponents: n x m = x n m . This property allows us to convert between radical notation and fractional exponents.
Converting the radical to a fractional exponent Applying this property to the given expression, we have 6 g 5 = g 6 5 .
Comparing with the options Now, we compare our result, g 6 5 , with the given options: A. g 5 6 B. 5 g 6 C. 6 5 g D. g 6 5 We see that option D matches our result.
Final Answer Therefore, the expression equivalent to 6 g 5 is g 6 5 .
Examples
Understanding fractional exponents is crucial in various fields, such as physics and engineering, where they are used to describe relationships between physical quantities. For example, the period of a simple pendulum is proportional to the square root of its length, which can be expressed using a fractional exponent. Similarly, in electrical engineering, the impedance of a capacitor is inversely proportional to the frequency raised to the power of 1/2. Mastering fractional exponents allows for accurate modeling and analysis of these phenomena.
