To simplify the expression [tex]\( x^{\frac{1}{3}} \left( x^{\frac{1}{2}} + 2 x^2 \right) \)[/tex], follow these steps:
1. Distribute [tex]\( x^{\frac{1}{3}} \)[/tex] across the terms inside the parenthesis:
[tex]\[
x^{\frac{1}{3}} \cdot x^{\frac{1}{2}} + x^{\frac{1}{3}} \cdot 2 x^2
\][/tex]
2. Simplify each term by using the property of exponents that [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex]:
[tex]\[
x^{\frac{1}{3} + \frac{1}{2}} + 2 x^{\frac{1}{3} + 2}
\][/tex]
3. Add the exponents in each term:
- For the first term: [tex]\( \frac{1}{3} + \frac{1}{2} \)[/tex].
To add these, you need a common denominator. The least common multiple of 3 and 2 is 6.
[tex]\[
\frac{1}{3} = \frac{2}{6}, \quad \frac{1}{2} = \frac{3}{6}
\][/tex]
[tex]\[
\frac{1}{3} + \frac{1}{2} = \frac{2}{6} + \frac{3}{6} = \frac{5}{6}
\][/tex]
- For the second term: [tex]\( \frac{1}{3} + 2 \)[/tex].
Express 2 as a fraction with the same denominator:
[tex]\[
2 = \frac{6}{3}
\][/tex]
[tex]\[
\frac{1}{3} + 2 = \frac{1}{3} + \frac{6}{3} = \frac{7}{3}
\][/tex]
4. Rewrite the expression with the simplified exponents:
[tex]\[
x^{\frac{5}{6}} + 2 x^{\frac{7}{3}}
\][/tex]
So the simplified expression for [tex]\( x^{\frac{1}{3}} \left( x^{\frac{1}{2}} + 2 x^2 \right) \)[/tex] is:
[tex]\[
x^{0.833333333333333} + 2 x^{2.33333333333333}
\][/tex]
