Question

1. what is the simplified form of ( x^{\frac{1}{3}} \times x^{\frac{1}{2}} )?

a. ( x^{\frac{1}{6}} )
b. ( x^{\frac{5}{6}} )
c. ( x^{\frac{1}{2}} )
d. ( x^{\frac{2}{5}} )

Explanation:

Step1: Recall exponent rule for multiplication

When multiplying exponents with the same base, we use the rule \( a^m \times a^n = a^{m + n} \). Here, the base is \( a \), \( m=\frac{1}{3} \), and \( n = \frac{1}{2} \).

Step2: Add the exponents

First, find a common denominator for \( \frac{1}{3} \) and \( \frac{1}{2} \), which is 6. Convert the fractions: \( \frac{1}{3}=\frac{2}{6} \) and \( \frac{1}{2}=\frac{3}{6} \). Then add them: \( \frac{2}{6}+\frac{3}{6}=\frac{5}{6} \). So \( a^{\frac{1}{3}} \times a^{\frac{1}{2}}=a^{\frac{1}{3}+\frac{1}{2}} = a^{\frac{5}{6}} \) (assuming option c is \( a^{\frac{5}{6}} \), maybe a typo in the original with "a⁵⁄₆" as the option).

Answer:

c. \( a^{\frac{5}{6}} \) (assuming the option c is intended to be \( a^{\frac{5}{6}} \) based on exponent multiplication rule)