To solve the expression [tex]\(\sqrt[3]{4} \cdot 4^{\frac{1}{3}} + \sqrt[5]{4} \cdot 4^{\frac{3}{5}}\)[/tex], we can follow these detailed steps:
Step 1: Simplify each term separately
1. First Term: [tex]\(\sqrt[3]{4} \cdot 4^{\frac{1}{3}}\)[/tex]
We know that [tex]\(\sqrt[3]{4}\)[/tex] can be written as [tex]\(4^{\frac{1}{3}}\)[/tex]. So the first term is:
[tex]\[
\sqrt[3]{4} \cdot 4^{\frac{1}{3}} = 4^{\frac{1}{3}} \cdot 4^{\frac{1}{3}}
\][/tex]
Using the properties of exponents, particularly that [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
[tex]\[
4^{\frac{1}{3}} \cdot 4^{\frac{1}{3}} = 4^{\frac{1}{3} + \frac{1}{3}} = 4^{\frac{2}{3}}
\][/tex]
Numerically, [tex]\(4^{\frac{2}{3}}\)[/tex] is approximately:
[tex]\[
4^{\frac{2}{3}} \approx 2.519842099789746
\][/tex]
2. Second Term: [tex]\(\sqrt[5]{4} \cdot 4^{\frac{3}{5}}\)[/tex]
Similarly, [tex]\(\sqrt[5]{4}\)[/tex] can be written as [tex]\(4^{\frac{1}{5}}\)[/tex]. Hence, the second term is:
[tex]\[
\sqrt[5]{4} \cdot 4^{\frac{3}{5}} = 4^{\frac{1}{5}} \cdot 4^{\frac{3}{5}}
\][/tex]
Using the properties of exponents:
[tex]\[
4^{\frac{1}{5}} \cdot 4^{\frac{3}{5}} = 4^{\frac{1}{5} + \frac{3}{5}} = 4^{\frac{4}{5}}
\][/tex]
Numerically, [tex]\(4^{\frac{4}{5}}\)[/tex] is approximately:
[tex]\[
4^{\frac{4}{5}} \approx 3.0314331330207955
\][/tex]
Step 2: Add the two simplified terms together
Now that we have the approximate values for each term, we can add them together:
[tex]\[
4^{\frac{2}{3}} + 4^{\frac{4}{5}} \approx 2.519842099789746 + 3.0314331330207955
\][/tex]
The sum of these values is approximately:
[tex]\[
2.519842099789746 + 3.0314331330207955 \approx 5.551275232810541
\][/tex]
So, the result of the expression [tex]\(\sqrt[3]{4} \cdot 4^{\frac{1}{3}} + \sqrt[5]{4} \cdot 4^{\frac{3}{5}}\)[/tex] is approximately [tex]\(5.551275232810541\)[/tex].
