Algebra Examples

$(7 - \sqrt{2}) (8 + \sqrt{2})$

Step 1

Expand $(7 - \sqrt{2}) (8 + \sqrt{2})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$7 (8 + \sqrt{2}) - \sqrt{2} (8 + \sqrt{2})$

Step 1.2

Apply the distributive property.

$7 \cdot 8 + 7 \sqrt{2} - \sqrt{2} (8 + \sqrt{2})$

Step 1.3

Apply the distributive property.

$7 \cdot 8 + 7 \sqrt{2} - \sqrt{2} \cdot 8 - \sqrt{2} \sqrt{2}$

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 $7$ by $8$ .

$56 + 7 \sqrt{2} - \sqrt{2} \cdot 8 - \sqrt{2} \sqrt{2}$

Step 2.1.2

Multiply $8$ by $- 1$ .

$56 + 7 \sqrt{2} - 8 \sqrt{2} - \sqrt{2} \sqrt{2}$

Step 2.1.3

Multiply $- \sqrt{2} \sqrt{2}$ .

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Step 2.1.3.1

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

$56 + 7 \sqrt{2} - 8 \sqrt{2} - ({\sqrt{2}}^{1} \sqrt{2})$

Step 2.1.3.2

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

$56 + 7 \sqrt{2} - 8 \sqrt{2} - ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

Step 2.1.3.3

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

$56 + 7 \sqrt{2} - 8 \sqrt{2} - {\sqrt{2}}^{1 + 1}$

Step 2.1.3.4

Add $1$ and $1$ .

$56 + 7 \sqrt{2} - 8 \sqrt{2} - {\sqrt{2}}^{2}$

Step 2.1.4

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

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Step 2.1.4.1

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

$56 + 7 \sqrt{2} - 8 \sqrt{2} - {(2^{\frac{1}{2}})}^{2}$

Step 2.1.4.2

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

$56 + 7 \sqrt{2} - 8 \sqrt{2} - 2^{\frac{1}{2} \cdot 2}$

Step 2.1.4.3

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

$56 + 7 \sqrt{2} - 8 \sqrt{2} - 2^{\frac{2}{2}}$

Step 2.1.4.4

Cancel the common factor of $2$ .

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Step 2.1.4.4.1

Cancel the common factor.

$56 + 7 \sqrt{2} - 8 \sqrt{2} - 2^{\frac{2}{2}}$

Step 2.1.4.4.2

Rewrite the expression.

$56 + 7 \sqrt{2} - 8 \sqrt{2} - 2^{1}$

Step 2.1.4.5

Evaluate the exponent.

$56 + 7 \sqrt{2} - 8 \sqrt{2} - 1 \cdot 2$

Step 2.1.5

Multiply $- 1$ by $2$ .

$56 + 7 \sqrt{2} - 8 \sqrt{2} - 2$

Step 2.2

Subtract $2$ from $56$ .

$54 + 7 \sqrt{2} - 8 \sqrt{2}$

Step 2.3

Subtract $8 \sqrt{2}$ from $7 \sqrt{2}$ .

$54 - \sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$54 - \sqrt{2}$

Decimal Form:

$52.58578643 \dots$