Simplify x^(1/3)(x^(1/2)+2x^2)

$x^{\frac{1}{3}} (x^{\frac{1}{2}} + 2 x^{2})$

Step 1

Apply the distributive property.

$x^{\frac{1}{3}} x^{\frac{1}{2}} + x^{\frac{1}{3}} (2 x^{2})$

Step 2

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{\frac{1}{3} + \frac{1}{2}} + x^{\frac{1}{3}} (2 x^{2})$

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{\frac{1}{3} \cdot \frac{2}{2} + \frac{1}{2}} + x^{\frac{1}{3}} (2 x^{2})$

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x^{\frac{1}{3} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{3}{3}} + x^{\frac{1}{3}} (2 x^{2})$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

$x^{\frac{2}{3 \cdot 2} + \frac{1}{2} \cdot \frac{3}{3}} + x^{\frac{1}{3}} (2 x^{2})$

Multiply $3$ by $2$ .

$x^{\frac{2}{6} + \frac{1}{2} \cdot \frac{3}{3}} + x^{\frac{1}{3}} (2 x^{2})$

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$x^{\frac{2}{6} + \frac{3}{2 \cdot 3}} + x^{\frac{1}{3}} (2 x^{2})$

Multiply $2$ by $3$ .

$x^{\frac{2}{6} + \frac{3}{6}} + x^{\frac{1}{3}} (2 x^{2})$

Combine the numerators over the common denominator.

$x^{\frac{2 + 3}{6}} + x^{\frac{1}{3}} (2 x^{2})$

Add $2$ and $3$ .

$x^{\frac{5}{6}} + x^{\frac{1}{3}} (2 x^{2})$

Step 3

Rewrite using the commutative property of multiplication.

$x^{\frac{5}{6}} + 2 x^{\frac{1}{3}} x^{2}$

Step 4

Multiply $x^{\frac{1}{3}}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$x^{\frac{5}{6}} + 2 (x^{2} x^{\frac{1}{3}})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{\frac{5}{6}} + 2 x^{2 + \frac{1}{3}}$

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x^{\frac{5}{6}} + 2 x^{2 \cdot \frac{3}{3} + \frac{1}{3}}$

Combine $2$ and $\frac{3}{3}$ .

$x^{\frac{5}{6}} + 2 x^{\frac{2 \cdot 3}{3} + \frac{1}{3}}$

Combine the numerators over the common denominator.

$x^{\frac{5}{6}} + 2 x^{\frac{2 \cdot 3 + 1}{3}}$

Simplify the numerator.

Multiply $2$ by $3$ .

$x^{\frac{5}{6}} + 2 x^{\frac{6 + 1}{3}}$

Add $6$ and $1$ .

$x^{\frac{5}{6}} + 2 x^{\frac{7}{3}}$