Questions: Simplify the expression to a single power of a. (a^(4/5) / a^(1/3))^(1/2) a^(3/8) a^(7/30) a^(1/2) a^(9/22)

Solution Steps

To simplify the expression \(\left(\frac{a^{\frac{4}{5}}}{a^{\frac{1}{3}}}\right)^{\frac{1}{2}}\) to a single power of \(a\), follow these steps:

1. Use the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\) to simplify the fraction inside the parentheses.
2. Apply the power of a power property \((a^m)^n = a^{m \cdot n}\) to simplify the expression further.

We start with the expression:

\[ \left(\frac{a^{\frac{4}{5}}}{a^{\frac{1}{3}}}\right)^{\frac{1}{2}} \]

Using the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\), we simplify the fraction inside the parentheses:

\[ \frac{4}{5} - \frac{1}{3} = \frac{12}{15} - \frac{5}{15} = \frac{7}{15} \]

Thus, we have:

\[ \left(a^{\frac{7}{15}}\right)^{\frac{1}{2}} \]

Next, we apply the power of a power property \((a^m)^n = a^{m \cdot n}\):

\[ \left(a^{\frac{7}{15}}\right)^{\frac{1}{2}} = a^{\frac{7}{15} \cdot \frac{1}{2}} = a^{\frac{7}{30}} \]

Final Answer