Certainly! Let's go through the steps to simplify the expression \(\sqrt[5]{4} \cdot \sqrt{2}\).
### Step 1: Rewrite in terms of rational exponents
To start, we will rewrite the radicals in terms of rational exponents:
[tex]\[
\sqrt[5]{4} = 4^{\frac{1}{5}}
\][/tex]
[tex]\[
\sqrt{2} = 2^{\frac{1}{2}}
\][/tex]
So the expression \(\sqrt[5]{4} \cdot \sqrt{2}\) can be written as:
[tex]\[
4^{\frac{1}{5}} \cdot 2^{\frac{1}{2}}
\][/tex]
### Step 2: Break down the base 4 in terms of base 2
Next, we rewrite \(4\) in terms of \(2\) since \(4\) is \(2^2\):
[tex]\[
4^{\frac{1}{5}} = (2^2)^{\frac{1}{5}} = 2^{2 \cdot \frac{1}{5}} = 2^{\frac{2}{5}}
\][/tex]
So the expression now is:
[tex]\[
2^{\frac{2}{5}} \cdot 2^{\frac{1}{2}}
\][/tex]
### Step 3: Use properties of exponents to combine the terms
With a common base, we can add the exponents:
[tex]\[
2^{\frac{2}{5}} \cdot 2^{\frac{1}{2}} = 2^{\frac{2}{5} + \frac{1}{2}}
\][/tex]
### Step 4: Find a common denominator and add the exponents
To add \(\frac{2}{5}\) and \(\frac{1}{2}\), we find a common denominator, which is 10:
[tex]\[
\frac{2}{5} = \frac{2 \cdot 2}{5 \cdot 2} = \frac{4}{10}
\][/tex]
[tex]\[
\frac{1}{2} = \frac{1 \cdot 5}{2 \cdot 5} = \frac{5}{10}
\][/tex]
Now add the fractions:
[tex]\[
\frac{4}{10} + \frac{5}{10} = \frac{4 + 5}{10} = \frac{9}{10}
\][/tex]
So we get:
[tex]\[
2^{\frac{9}{10}}
\][/tex]
### Step 5: Evaluate the result numerically
Finally, \(2^{\frac{9}{10}}\) evaluates to approximately \(2.5198420997897464\).
### Summary:
Thus, the simplified form of \(\sqrt[5]{4} \cdot \sqrt{2}\) using rational exponents and properties of exponents is \(2^{\frac{9}{10}}\), which numerically evaluates to:
[tex]\[
2.5198420997897464
\][/tex]
The intermediate steps are also confirmed:
- \(4^{\frac{1}{6}} \approx 1.2599210498948732\)
- \(2^{\frac{1}{1}} = 2.0\)
So, the expression can be simplified and the resulting value is [tex]\( \boxed{2.5198420997897464} \)[/tex].
