To simplify the expression [tex]\( x^{\frac{1}{3}}\left(x^{\frac{1}{2}} + 2x^2\right) \)[/tex], follow these steps:
1. Distribute [tex]\( x^{\frac{1}{3}} \)[/tex] across the terms inside the parentheses:
[tex]\[
x^{\frac{1}{3}} \cdot x^{\frac{1}{2}} + x^{\frac{1}{3}} \cdot 2x^2
\][/tex]
2. Simplify each term by adding the exponents (since the bases are the same):
- For the first term [tex]\( x^{\frac{1}{3}} \cdot x^{\frac{1}{2}} \)[/tex]:
[tex]\[
x^{\frac{1}{3} + \frac{1}{2}}
\][/tex]
Add the exponents:
[tex]\[
\frac{1}{3} + \frac{1}{2} = \frac{2}{6} + \frac{3}{6} = \frac{5}{6}
\][/tex]
So the first term simplifies to:
[tex]\[
x^{\frac{5}{6}}
\][/tex]
- For the second term [tex]\( x^{\frac{1}{3}} \cdot 2x^2 \)[/tex]:
[tex]\[
2 \cdot x^{\frac{1}{3} + 2}
\][/tex]
Add the exponents:
[tex]\[
\frac{1}{3} + 2 = \frac{1}{3} + \frac{6}{3} = \frac{7}{3}
\][/tex]
So the second term simplifies to:
[tex]\[
2x^{\frac{7}{3}}
\][/tex]
3. Combine the simplified terms:
[tex]\[
x^{\frac{5}{6}} + 2x^{\frac{7}{3}}
\][/tex]
Thus, the simplified expression is:
[tex]\[
x^{\frac{5}{6}} + 2x^{\frac{7}{3}}
\][/tex]
