To simplify the expression \(7^{2/3} \cdot 7^{1/5}\), we can use the properties of exponents. Specifically, we will use the property that states:
[tex]\[
a^m \cdot a^n = a^{m+n}
\][/tex]
In this situation, \(a\) is 7, \(m\) is \(\frac{2}{3}\), and \(n\) is \(\frac{1}{5}\).
1. First, identify the exponents:
[tex]\[
m = \frac{2}{3}, \quad n = \frac{1}{5}
\][/tex]
2. Add the exponents together:
[tex]\[
\frac{2}{3} + \frac{1}{5}
\][/tex]
To add these fractions, find a common denominator. The denominators are 3 and 5, and a common denominator is 15. Convert each fraction:
[tex]\[
\frac{2}{3} = \frac{2 \cdot 5}{3 \cdot 5} = \frac{10}{15}
\][/tex]
[tex]\[
\frac{1}{5} = \frac{1 \cdot 3}{5 \cdot 3} = \frac{3}{15}
\][/tex]
Now, add the fractions:
[tex]\[
\frac{10}{15} + \frac{3}{15} = \frac{10 + 3}{15} = \frac{13}{15}
\][/tex]
3. Rewrite the original expression using the sum of the exponents:
[tex]\[
7^{2/3} \cdot 7^{1/5} = 7^{(2/3 + 1/5)} = 7^{13/15}
\][/tex]
4. Evaluate \(7^{13/15}\). The numerical value of this expression is approximately:
[tex]\[
7^{13/15} \approx 5.400305156921598
\][/tex]
Thus, the simplified form of the expression [tex]\(7^{2/3} \cdot 7^{1/5}\)[/tex] is [tex]\(7^{13/15}\)[/tex], and its approximate numerical value is 5.400305156921598.
