To simplify the expression [tex]\( 4^{\frac{1}{3}} \cdot 4^{\frac{1}{5}} \)[/tex], we can use the product of powers property, which states that [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex].
1. Identify the base and the exponents in the expression:
- The base is [tex]\( 4 \)[/tex].
- The exponents are [tex]\( \frac{1}{3} \)[/tex] and [tex]\( \frac{1}{5} \)[/tex].
2. Using the product of powers property, combine the exponents:
[tex]\[
\frac{1}{3} + \frac{1}{5}
\][/tex]
3. To add these fractions, find a common denominator. The common denominator of 3 and 5 is 15:
[tex]\[
\frac{1}{3} = \frac{5}{15} \quad \text{and} \quad \frac{1}{5} = \frac{3}{15}
\][/tex]
4. Add the fractions:
[tex]\[
\frac{5}{15} + \frac{3}{15} = \frac{8}{15}
\][/tex]
5. Now the expression becomes:
[tex]\[
4^{\frac{1}{3}} \cdot 4^{\frac{1}{5}} = 4^{\frac{8}{15}}
\][/tex]
6. Finally, calculate [tex]\( 4^{\frac{8}{15}} \)[/tex]:
[tex]\[
4^{\frac{8}{15}} \approx 2.0945882456412535
\][/tex]
Therefore, [tex]\( 4^{\frac{1}{3}} \cdot 4^{\frac{1}{5}} \approx 2.0945882456412535 \)[/tex].
