Calculus Examples

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

Step 1

Expand $(\sqrt{x} - 1) (\sqrt{x} + 1)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $\sqrt{x} \sqrt{x} + \sqrt{x} \cdot 1 - 1 \sqrt{x} - 1 \cdot 1$ .

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Step 2.1.1

Reorder the factors in the terms $\sqrt{x} \cdot 1$ and $- 1 \sqrt{x}$ .

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

Step 2.1.2

Subtract $1 \sqrt{x}$ from $1 \sqrt{x}$ .

$\sqrt{x} \sqrt{x} + 0 - 1 \cdot 1$

Step 2.1.3

Add $\sqrt{x} \sqrt{x}$ and $0$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.1.4

Add $1$ and $1$ .

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.2.4

Cancel the common factor of $2$ .

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Step 2.2.2.4.1

Cancel the common factor.

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

Step 2.2.2.4.2

Rewrite the expression.

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

Step 2.2.2.5

Simplify.

$x - 1 \cdot 1$

Step 2.2.3

Multiply $- 1$ by $1$ .

$x - 1$