Algebra Examples

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

Step 1

Rewrite ${(\sqrt{x} - x^{2})}^{2}$ as $(\sqrt{x} - x^{2}) (\sqrt{x} - x^{2})$ .

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

Step 2

Expand $(\sqrt{x} - x^{2}) (\sqrt{x} - x^{2})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

$\sqrt{x} \sqrt{x} + \sqrt{x} (- x^{2}) - x^{2} (\sqrt{x} - x^{2})$

Step 2.3

Apply the distributive property.

$\sqrt{x} \sqrt{x} + \sqrt{x} (- x^{2}) - x^{2} \sqrt{x} - x^{2} (- x^{2})$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $\sqrt{x} \sqrt{x}$ .

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Step 3.1.1.1

Raise $\sqrt{x}$ to the power of $1$ .

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

Step 3.1.1.2

Raise $\sqrt{x}$ to the power of $1$ .

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

Step 3.1.1.3

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

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

Step 3.1.1.4

Add $1$ and $1$ .

${\sqrt{x}}^{2} + \sqrt{x} (- x^{2}) - x^{2} \sqrt{x} - x^{2} (- x^{2})$

Step 3.1.2

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 3.1.2.4

Cancel the common factor of $2$ .

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Step 3.1.2.4.1

Cancel the common factor.

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

Step 3.1.2.4.2

Rewrite the expression.

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

Step 3.1.2.5

Simplify.

$x + \sqrt{x} (- x^{2}) - x^{2} \sqrt{x} - x^{2} (- x^{2})$

Step 3.1.3

Rewrite using the commutative property of multiplication.

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

Step 3.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.1.5

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

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Step 3.1.5.1

Move $x^{2}$ .

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

Step 3.1.5.2

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

$x - \sqrt{x} x^{2} - x^{2} \sqrt{x} - 1 \cdot - 1 x^{2 + 2}$

Step 3.1.5.3

Add $2$ and $2$ .

$x - \sqrt{x} x^{2} - x^{2} \sqrt{x} - 1 \cdot - 1 x^{4}$

Step 3.1.6

Multiply $- 1$ by $- 1$ .

$x - \sqrt{x} x^{2} - x^{2} \sqrt{x} + 1 x^{4}$

Step 3.1.7

Multiply $x^{4}$ by $1$ .

$x - \sqrt{x} x^{2} - x^{2} \sqrt{x} + x^{4}$

Step 3.2

Reorder the factors of $- \sqrt{x} x^{2}$ .

$x - x^{2} \sqrt{x} - x^{2} \sqrt{x} + x^{4}$

Step 3.3

Subtract $x^{2} \sqrt{x}$ from $- x^{2} \sqrt{x}$ .

$x - 2 x^{2} \sqrt{x} + x^{4}$