Precalculus Examples

$(\sqrt{3} - 1) (1 - \sqrt{3})$

Step 1

Expand $(\sqrt{3} - 1) (1 - \sqrt{3})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$\sqrt{3} (1 - \sqrt{3}) - 1 (1 - \sqrt{3})$

Step 1.2

Apply the distributive property.

$\sqrt{3} \cdot 1 + \sqrt{3} (- \sqrt{3}) - 1 (1 - \sqrt{3})$

Step 1.3

Apply the distributive property.

$\sqrt{3} \cdot 1 + \sqrt{3} (- \sqrt{3}) - 1 \cdot 1 - 1 (- \sqrt{3})$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $\sqrt{3}$ by $1$ .

$\sqrt{3} + \sqrt{3} (- \sqrt{3}) - 1 \cdot 1 - 1 (- \sqrt{3})$

Step 2.1.2

Multiply $\sqrt{3} (- \sqrt{3})$ .

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Step 2.1.2.1

Raise $\sqrt{3}$ to the power of $1$ .

$\sqrt{3} - ({\sqrt{3}}^{1} \sqrt{3}) - 1 \cdot 1 - 1 (- \sqrt{3})$

Step 2.1.2.2

Raise $\sqrt{3}$ to the power of $1$ .

$\sqrt{3} - ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) - 1 \cdot 1 - 1 (- \sqrt{3})$

Step 2.1.2.3

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

$\sqrt{3} - {\sqrt{3}}^{1 + 1} - 1 \cdot 1 - 1 (- \sqrt{3})$

Step 2.1.2.4

Add $1$ and $1$ .

$\sqrt{3} - {\sqrt{3}}^{2} - 1 \cdot 1 - 1 (- \sqrt{3})$

Step 2.1.3

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

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

Step 2.1.3.4

Cancel the common factor of $2$ .

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Step 2.1.3.4.1

Cancel the common factor.

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

Step 2.1.3.4.2

Rewrite the expression.

$\sqrt{3} - 3^{1} - 1 \cdot 1 - 1 (- \sqrt{3})$

Step 2.1.3.5

Evaluate the exponent.

$\sqrt{3} - 1 \cdot 3 - 1 \cdot 1 - 1 (- \sqrt{3})$

Step 2.1.4

Multiply $- 1$ by $3$ .

$\sqrt{3} - 3 - 1 \cdot 1 - 1 (- \sqrt{3})$

Step 2.1.5

Multiply $- 1$ by $1$ .

$\sqrt{3} - 3 - 1 - 1 (- \sqrt{3})$

Step 2.1.6

Multiply $- 1 (- \sqrt{3})$ .

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Step 2.1.6.1

Multiply $- 1$ by $- 1$ .

$\sqrt{3} - 3 - 1 + 1 \sqrt{3}$

Step 2.1.6.2

Multiply $\sqrt{3}$ by $1$ .

$\sqrt{3} - 3 - 1 + \sqrt{3}$

Step 2.2

Add $\sqrt{3}$ and $\sqrt{3}$ .

$- 3 - 1 + 2 \sqrt{3}$

Step 2.3

Subtract $1$ from $- 3$ .

$- 4 + 2 \sqrt{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- 4 + 2 \sqrt{3}$

Decimal Form:

$- 0.53589838 \dots$