To simplify the expression [tex]\(\sqrt[5]{4} \cdot \sqrt{2}\)[/tex], let's follow these steps comprehensively.
### Step 1: Rewrite Using Rational Exponents
Given:
[tex]\[ \sqrt[5]{4} \cdot \sqrt{2} \][/tex]
Let's express each term using rational exponents.
1. [tex]\(\sqrt[5]{4}\)[/tex] can be written as:
[tex]\[ 4^{\frac{1}{5}} \][/tex]
2. [tex]\(\sqrt{2}\)[/tex] can be written as:
[tex]\[ 2^{\frac{1}{2}} \][/tex]
Thus, the expression becomes:
[tex]\[ 4^{\frac{1}{5}} \cdot 2^{\frac{1}{2}} \][/tex]
### Step 2: Rewrite 4 as a Power of 2
Notice that [tex]\(4\)[/tex] can be expressed as [tex]\(2^2\)[/tex]. Therefore, we can rewrite [tex]\(4^{\frac{1}{5}}\)[/tex] as:
[tex]\[ \left(2^2\right)^{\frac{1}{5}} \][/tex]
Using the properties of exponents [tex]\((a^m)^n = a^{mn}\)[/tex], we get:
[tex]\[ 2^{\frac{2}{5}} \][/tex]
Now the expression is:
[tex]\[ 2^{\frac{2}{5}} \cdot 2^{\frac{1}{2}} \][/tex]
### Step 3: Combine the Exponents
Since the bases are the same (base 2), 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: Simplify the Exponent
To add the exponents [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{1}{2}\)[/tex], we need a common denominator. The least common denominator of 5 and 2 is 10:
Convert [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{1}{2}\)[/tex] to equivalent fractions with a denominator of 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]\[ \frac{4}{10} + \frac{5}{10} = \frac{9}{10} \][/tex]
Thus, our expression with the combined exponent is:
[tex]\[ 2^{\frac{9}{10}} \][/tex]
### Step 5: Simplified Form
The simplified form of the original expression [tex]\(\sqrt[5]{4} \cdot \sqrt{2}\)[/tex] is:
[tex]\[ 2^{\frac{9}{10}} \][/tex]
### Step 6: Approximate Numerical Value
To provide the approximate numerical value, we compute:
[tex]\[ 2^{\frac{9}{10}} \approx 1.8660659830736148 \][/tex]
Thus, the simplified expression [tex]\(\sqrt[5]{4} \cdot \sqrt{2}\)[/tex] in its exact rational exponent form is [tex]\(2^{\frac{9}{10}}\)[/tex], and the approximate numerical value is [tex]\(1.8660659830736148\)[/tex].
