Algebra Examples

$x^{2} + \sqrt{x}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$x^{2} + x^{\frac{1}{2}}$

Step 2

Factor $x^{\frac{1}{2}}$ out of $x^{2} + x^{\frac{1}{2}}$ .

Tap for more steps...

Step 2.1

Factor $x^{\frac{1}{2}}$ out of $x^{2}$ .

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

Step 2.2

Multiply by $1$ .

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

Step 2.3

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} x^{\frac{3}{2}} + x^{\frac{1}{2}} \cdot 1$ .

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

Step 3

Rewrite $x^{\frac{3}{2}}$ as ${(x^{\frac{1}{2}})}^{3}$ .

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

Step 4

Rewrite $1$ as $1^{3}$ .

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

Step 5

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x^{\frac{1}{2}}$ and $b = 1$ .

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

Step 6

Factor.

Tap for more steps...

Step 6.1

Simplify.

Tap for more steps...

Step 6.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 6.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 6.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.1.1.2.1

Cancel the common factor.

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

Step 6.1.1.2.2

Rewrite the expression.

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

Step 6.1.2

Simplify.

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

Step 6.1.3

Multiply $- 1$ by $1$ .

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

Step 6.1.4

One to any power is one.

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

Step 6.2

Remove unnecessary parentheses.

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