Sure, let's simplify the given expression using the quotient of powers property. The expression we need to simplify is:
[tex]\[
\frac{2^{\frac{2}{5}}}{2^{\frac{1}{10}}}
\][/tex]
### Step-by-Step Solution
1. Identify the exponents:
The numerator has an exponent of [tex]\(\frac{2}{5}\)[/tex].
The denominator has an exponent of [tex]\(\frac{1}{10}\)[/tex].
2. Apply the quotient of powers property:
The quotient of powers property states that [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]. Therefore, we subtract the exponent in the denominator from the exponent in the numerator.
[tex]\[
\frac{2^{\frac{2}{5}}}{2^{\frac{1}{10}}} = 2^{\frac{2}{5} - \frac{1}{10}}
\][/tex]
3. Common denominator for exponents:
To subtract the exponents, we need a common denominator. The denominators are 5 and 10. The least common denominator is 10.
[tex]\[
\frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10}
\][/tex]
Now we have:
[tex]\[
2^{\frac{4}{10} - \frac{1}{10}}
\][/tex]
4. Subtract the exponents:
Subtracting the exponents with a common denominator:
[tex]\[
\frac{4}{10} - \frac{1}{10} = \frac{4 - 1}{10} = \frac{3}{10}
\][/tex]
Thus, we get:
[tex]\[
2^{\frac{3}{10}}
\][/tex]
5. Simplify the expression:
So, the simplified form of the original expression is:
[tex]\[
2^{\frac{3}{10}}
\][/tex]
### Calculation of the Result
For completeness, let's also express the actual value:
[tex]\[
2^{\frac{3}{10}} \approx 1.2311444133449163
\][/tex]
So, [tex]\(\frac{2^{\frac{2}{5}}}{2^{\frac{1}{10}}} = 2^{\frac{3}{10}}\approx 1.2311444133449163 \)[/tex].
But for the exact answer in the simplified form:
[tex]\[
\boxed{2^{\frac{3}{10}}}
\][/tex]
