Pre-Algebra Examples

$(9 + \sqrt{3}) (9 - \sqrt{3})$

Step 1

Expand $(9 + \sqrt{3}) (9 - \sqrt{3})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$9 (9 - \sqrt{3}) + \sqrt{3} (9 - \sqrt{3})$

Step 1.2

Apply the distributive property.

$9 \cdot 9 + 9 (- \sqrt{3}) + \sqrt{3} (9 - \sqrt{3})$

Step 1.3

Apply the distributive property.

$9 \cdot 9 + 9 (- \sqrt{3}) + \sqrt{3} \cdot 9 + \sqrt{3} (- \sqrt{3})$

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 $9$ by $9$ .

$81 + 9 (- \sqrt{3}) + \sqrt{3} \cdot 9 + \sqrt{3} (- \sqrt{3})$

Step 2.1.2

Multiply $- 1$ by $9$ .

$81 - 9 \sqrt{3} + \sqrt{3} \cdot 9 + \sqrt{3} (- \sqrt{3})$

Step 2.1.3

Move $9$ to the left of $\sqrt{3}$ .

$81 - 9 \sqrt{3} + 9 \cdot \sqrt{3} + \sqrt{3} (- \sqrt{3})$

Step 2.1.4

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

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Step 2.1.4.1

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

$81 - 9 \sqrt{3} + 9 \sqrt{3} - ({\sqrt{3}}^{1} \sqrt{3})$

Step 2.1.4.2

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

$81 - 9 \sqrt{3} + 9 \sqrt{3} - ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 2.1.4.3

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

$81 - 9 \sqrt{3} + 9 \sqrt{3} - {\sqrt{3}}^{1 + 1}$

Step 2.1.4.4

Add $1$ and $1$ .

$81 - 9 \sqrt{3} + 9 \sqrt{3} - {\sqrt{3}}^{2}$

Step 2.1.5

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

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Step 2.1.5.1

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

$81 - 9 \sqrt{3} + 9 \sqrt{3} - {(3^{\frac{1}{2}})}^{2}$

Step 2.1.5.2

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

$81 - 9 \sqrt{3} + 9 \sqrt{3} - 3^{\frac{1}{2} \cdot 2}$

Step 2.1.5.3

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

$81 - 9 \sqrt{3} + 9 \sqrt{3} - 3^{\frac{2}{2}}$

Step 2.1.5.4

Cancel the common factor of $2$ .

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Step 2.1.5.4.1

Cancel the common factor.

$81 - 9 \sqrt{3} + 9 \sqrt{3} - 3^{\frac{2}{2}}$

Step 2.1.5.4.2

Rewrite the expression.

$81 - 9 \sqrt{3} + 9 \sqrt{3} - 3^{1}$

Step 2.1.5.5

Evaluate the exponent.

$81 - 9 \sqrt{3} + 9 \sqrt{3} - 1 \cdot 3$

Step 2.1.6

Multiply $- 1$ by $3$ .

$81 - 9 \sqrt{3} + 9 \sqrt{3} - 3$

Step 2.2

Subtract $3$ from $81$ .

$78 - 9 \sqrt{3} + 9 \sqrt{3}$

Step 2.3

Add $- 9 \sqrt{3}$ and $9 \sqrt{3}$ .

$78 + 0$

Step 2.4

Add $78$ and $0$ .

$78$