Question

use the rules for exponents to simplify the following expression. write the final answers in exponential form when necessary. assume that all variables represent non - negative values.\\(x^{\frac{1}{2}}\cdot x^{\frac{1}{7}}\\)\\(x^{\frac{1}{2}}\cdot x^{\frac{1}{7}}=\square\\)\\(\text{(simplify your answer. use integers or fractions for any numbers in the expression. type your answer using exponential notation.)}\\)

Explanation:

Step1: Recall exponent rule for multiplication

When multiplying two exponential expressions with the same base, we add the exponents: \(a^m \cdot a^n = a^{m + n}\). Here, the base is \(x\), \(m=\frac{1}{2}\), and \(n = \frac{1}{7}\).

Step2: Add the exponents

Calculate \(\frac{1}{2}+\frac{1}{7}\). Find a common denominator, which is \(14\). So \(\frac{1}{2}=\frac{7}{14}\) and \(\frac{1}{7}=\frac{2}{14}\). Then \(\frac{7}{14}+\frac{2}{14}=\frac{7 + 2}{14}=\frac{9}{14}\). Thus, \(x^{\frac{1}{2}}\cdot x^{\frac{1}{7}}=x^{\frac{9}{14}}\).

Answer:

\(x^{\frac{9}{14}}\)