Factor x^2- square root of x

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

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 $x^{2} - x^{\frac{1}{2}}$ .

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

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

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

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

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 difference 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.

Simplify.

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Multiply $x^{\frac{1}{2}}$ by $1$ .

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

One to any power is one.

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

Remove unnecessary parentheses.

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

Factor x^2- square root of x

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