Algebra Examples

$(7 - \sqrt{7}) (- 6 + \sqrt{7})$

Step 1

Expand $(7 - \sqrt{7}) (- 6 + \sqrt{7})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$7 (- 6 + \sqrt{7}) - \sqrt{7} (- 6 + \sqrt{7})$

Step 1.2

Apply the distributive property.

$7 \cdot - 6 + 7 \sqrt{7} - \sqrt{7} (- 6 + \sqrt{7})$

Step 1.3

Apply the distributive property.

$7 \cdot - 6 + 7 \sqrt{7} - \sqrt{7} \cdot - 6 - \sqrt{7} \sqrt{7}$

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 $- 6$ .

$- 42 + 7 \sqrt{7} - \sqrt{7} \cdot - 6 - \sqrt{7} \sqrt{7}$

Step 2.1.2

Multiply $- 6$ by $- 1$ .

$- 42 + 7 \sqrt{7} + 6 \sqrt{7} - \sqrt{7} \sqrt{7}$

Step 2.1.3

Multiply $- \sqrt{7} \sqrt{7}$ .

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Step 2.1.3.1

Raise $\sqrt{7}$ to the power of $1$ .

$- 42 + 7 \sqrt{7} + 6 \sqrt{7} - ({\sqrt{7}}^{1} \sqrt{7})$

Step 2.1.3.2

Raise $\sqrt{7}$ to the power of $1$ .

$- 42 + 7 \sqrt{7} + 6 \sqrt{7} - ({\sqrt{7}}^{1} {\sqrt{7}}^{1})$

Step 2.1.3.3

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

$- 42 + 7 \sqrt{7} + 6 \sqrt{7} - {\sqrt{7}}^{1 + 1}$

Step 2.1.3.4

Add $1$ and $1$ .

$- 42 + 7 \sqrt{7} + 6 \sqrt{7} - {\sqrt{7}}^{2}$

Step 2.1.4

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 2.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$- 42 + 7 \sqrt{7} + 6 \sqrt{7} - {(7^{\frac{1}{2}})}^{2}$

Step 2.1.4.2

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

$- 42 + 7 \sqrt{7} + 6 \sqrt{7} - 7^{\frac{1}{2} \cdot 2}$

Step 2.1.4.3

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

$- 42 + 7 \sqrt{7} + 6 \sqrt{7} - 7^{\frac{2}{2}}$

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.

$- 42 + 7 \sqrt{7} + 6 \sqrt{7} - 7^{\frac{2}{2}}$

Step 2.1.4.4.2

Rewrite the expression.

$- 42 + 7 \sqrt{7} + 6 \sqrt{7} - 7^{1}$

Step 2.1.4.5

Evaluate the exponent.

$- 42 + 7 \sqrt{7} + 6 \sqrt{7} - 1 \cdot 7$

Step 2.1.5

Multiply $- 1$ by $7$ .

$- 42 + 7 \sqrt{7} + 6 \sqrt{7} - 7$

Step 2.2

Subtract $7$ from $- 42$ .

$- 49 + 7 \sqrt{7} + 6 \sqrt{7}$

Step 2.3

Add $7 \sqrt{7}$ and $6 \sqrt{7}$ .

$- 49 + 13 \sqrt{7}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- 49 + 13 \sqrt{7}$

Decimal Form:

$- 14.60523295 \dots$