To solve the expression [tex]\(\sqrt{\frac{1}{2}} \times \sqrt[3]{x^2}\)[/tex], let's break it down into manageable parts:
1. Calculate the square root of [tex]\(\frac{1}{2}\)[/tex]:
The square root of [tex]\(\frac{1}{2}\)[/tex] is:
[tex]\[
\sqrt{\frac{1}{2}}
\][/tex]
The value of [tex]\(\frac{1}{2}\)[/tex] can be expressed as a decimal, which is 0.5. The square root of 0.5 is approximately:
[tex]\[
\sqrt{0.5} \approx 0.707106781186548
\][/tex]
2. Calculate the cube root of [tex]\(x^2\)[/tex]:
The cube root of [tex]\(x^2\)[/tex] can be written as:
[tex]\[
\sqrt[3]{x^2}
\][/tex]
Alternatively, we can express the cube root using exponents:
[tex]\[
\left( x^2 \right)^{\frac{1}{3}} = x^{2 \cdot \frac{1}{3}} = x^{\frac{2}{3}}
\][/tex]
So, [tex]\(\sqrt[3]{x^2}\)[/tex] is:
[tex]\[
x^{\frac{2}{3}}
\][/tex]
3. Combine the results:
Now we multiply the two parts together:
[tex]\[
\sqrt{\frac{1}{2}} \times \sqrt[3]{x^2} \approx 0.707106781186548 \times x^{\frac{2}{3}}
\][/tex]
Therefore, the expression [tex]\(\sqrt{\frac{1}{2}} \times \sqrt[3]{x^2}\)[/tex] simplifies to approximately:
[tex]\[
0.707106781186548 \times x^{0.666666666666667}
\][/tex]
Combining these, we get the final result:
[tex]\[
0.707106781186548 \cdot x^{0.666666666666667}
\][/tex]
Which is:
[tex]\[
0.707106781186548 x^{0.666666666666667}
\][/tex]
This is the simplified form of the given expression.
