Factor x square root of x-1

$x \sqrt{x} - 1$

Step 1

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

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

Step 2

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

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

Raise $x$ to the power of $1$ .

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

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

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $2$ and $1$ .

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

Step 3

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

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

Step 4

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

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

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 $b = 1$ .

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

Step 6

Simplify.

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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

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

One to any power is one.

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

Factor x square root of x-1

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