Algebra Examples

$(3 \sqrt{2} - 9) (3 \sqrt{2} + 9)$

Step 1

Expand $(3 \sqrt{2} - 9) (3 \sqrt{2} + 9)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$3 \sqrt{2} (3 \sqrt{2} + 9) - 9 (3 \sqrt{2} + 9)$

Step 1.2

Apply the distributive property.

$3 \sqrt{2} (3 \sqrt{2}) + 3 \sqrt{2} \cdot 9 - 9 (3 \sqrt{2} + 9)$

Step 1.3

Apply the distributive property.

$3 \sqrt{2} (3 \sqrt{2}) + 3 \sqrt{2} \cdot 9 - 9 (3 \sqrt{2}) - 9 \cdot 9$

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 $3 \sqrt{2} (3 \sqrt{2})$ .

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Step 2.1.1.1

Multiply $3$ by $3$ .

$9 \sqrt{2} \sqrt{2} + 3 \sqrt{2} \cdot 9 - 9 (3 \sqrt{2}) - 9 \cdot 9$

Step 2.1.1.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 2.1.1.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 2.1.1.4

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

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

Step 2.1.1.5

Add $1$ and $1$ .

$9 {\sqrt{2}}^{2} + 3 \sqrt{2} \cdot 9 - 9 (3 \sqrt{2}) - 9 \cdot 9$

Step 2.1.2

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

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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

Step 2.1.2.4

Cancel the common factor of $2$ .

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Step 2.1.2.4.1

Cancel the common factor.

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

Step 2.1.2.4.2

Rewrite the expression.

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

Step 2.1.2.5

Evaluate the exponent.

$9 \cdot 2 + 3 \sqrt{2} \cdot 9 - 9 (3 \sqrt{2}) - 9 \cdot 9$

Step 2.1.3

Multiply $9$ by $2$ .

$18 + 3 \sqrt{2} \cdot 9 - 9 (3 \sqrt{2}) - 9 \cdot 9$

Step 2.1.4

Multiply $9$ by $3$ .

$18 + 27 \sqrt{2} - 9 (3 \sqrt{2}) - 9 \cdot 9$

Step 2.1.5

Multiply $3$ by $- 9$ .

$18 + 27 \sqrt{2} - 27 \sqrt{2} - 9 \cdot 9$

Step 2.1.6

Multiply $- 9$ by $9$ .

$18 + 27 \sqrt{2} - 27 \sqrt{2} - 81$

Step 2.2

Subtract $81$ from $18$ .

$- 63 + 27 \sqrt{2} - 27 \sqrt{2}$

Step 2.3

Subtract $27 \sqrt{2}$ from $27 \sqrt{2}$ .

$- 63 + 0$

Step 2.4

Add $- 63$ and $0$ .

$- 63$