To simplify the given expression [tex]\(\frac{2^{\frac{2}{5}}}{2^{\frac{1}{10}}}\)[/tex], we will use the quotient of powers property:
### Step 1: Apply the Quotient of Powers Property
The quotient of powers property states:
[tex]\[
\frac{a^m}{a^n} = a^{m-n}
\][/tex]
In our expression, [tex]\(a = 2\)[/tex], [tex]\(m = \frac{2}{5}\)[/tex], and [tex]\(n = \frac{1}{10}\)[/tex]. So, we have:
[tex]\[
\frac{2^{\frac{2}{5}}}{2^{\frac{1}{10}}} = 2^{\left(\frac{2}{5} - \frac{1}{10}\right)}
\][/tex]
### Step 2: Simplify the Exponent
To simplify the exponent [tex]\(\frac{2}{5} - \frac{1}{10}\)[/tex], we need a common denominator.
[tex]\[
\frac{2}{5} = \frac{4}{10} \quad (\text{since} \ \frac{2}{5} \times \frac{2}{2} = \frac{4}{10})
\][/tex]
Now, subtract the exponents:
[tex]\[
\frac{4}{10} - \frac{1}{10} = \frac{3}{10}
\][/tex]
### Step 3: Simplify the Expression
Substitute the simplified exponent back into the expression:
[tex]\[
2^{\frac{3}{10}}
\][/tex]
### Step 4: Evaluate the Result
Finally, evaluate [tex]\(2^{\frac{3}{10}}\)[/tex] to get the numerical answer.
Thus, the simplified form of the given expression is:
[tex]\[
\boxed{2^{\frac{3}{10}} = 1.2311444133449163}
\][/tex]
