Algebra Examples

$\frac{7 \sqrt{6} + 6 \sqrt{7}}{7 \sqrt{7} + 6 \sqrt{6}}$

Step 1

Multiply $\frac{7 \sqrt{6} + 6 \sqrt{7}}{7 \sqrt{7} + 6 \sqrt{6}}$ by $\frac{7 \sqrt{7} - 6 \sqrt{6}}{7 \sqrt{7} - 6 \sqrt{6}}$ .

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

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{7 \sqrt{6} + 6 \sqrt{7}}{7 \sqrt{7} + 6 \sqrt{6}}$ by $\frac{7 \sqrt{7} - 6 \sqrt{6}}{7 \sqrt{7} - 6 \sqrt{6}}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(7 \sqrt{6} + 6 \sqrt{7}) (7 \sqrt{7} - 6 \sqrt{6})}{49 {\sqrt{7}}^{2} - 42 \sqrt{42} + 42 \sqrt{42} - 36 {\sqrt{6}}^{2}}$

Step 2.3

Simplify.

$\frac{(7 \sqrt{6} + 6 \sqrt{7}) (7 \sqrt{7} - 6 \sqrt{6})}{127}$

Step 3

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

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Step 3.1

Apply the distributive property.

$\frac{7 \sqrt{6} (7 \sqrt{7} - 6 \sqrt{6}) + 6 \sqrt{7} (7 \sqrt{7} - 6 \sqrt{6})}{127}$

Step 3.2

Apply the distributive property.

$\frac{7 \sqrt{6} (7 \sqrt{7}) + 7 \sqrt{6} (- 6 \sqrt{6}) + 6 \sqrt{7} (7 \sqrt{7} - 6 \sqrt{6})}{127}$

Step 3.3

Apply the distributive property.

$\frac{7 \sqrt{6} (7 \sqrt{7}) + 7 \sqrt{6} (- 6 \sqrt{6}) + 6 \sqrt{7} (7 \sqrt{7}) + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $7 \sqrt{6} (7 \sqrt{7})$ .

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Step 4.1.1.1

Multiply $7$ by $7$ .

$\frac{49 \sqrt{6} \sqrt{7} + 7 \sqrt{6} (- 6 \sqrt{6}) + 6 \sqrt{7} (7 \sqrt{7}) + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.1.2

Combine using the product rule for radicals.

$\frac{49 \sqrt{7 \cdot 6} + 7 \sqrt{6} (- 6 \sqrt{6}) + 6 \sqrt{7} (7 \sqrt{7}) + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.1.3

Multiply $7$ by $6$ .

$\frac{49 \sqrt{42} + 7 \sqrt{6} (- 6 \sqrt{6}) + 6 \sqrt{7} (7 \sqrt{7}) + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.2

Multiply $7 \sqrt{6} (- 6 \sqrt{6})$ .

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Step 4.1.2.1

Multiply $- 6$ by $7$ .

$\frac{49 \sqrt{42} - 42 \sqrt{6} \sqrt{6} + 6 \sqrt{7} (7 \sqrt{7}) + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.2.2

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{49 \sqrt{42} - 42 ({\sqrt{6}}^{1} \sqrt{6}) + 6 \sqrt{7} (7 \sqrt{7}) + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.2.3

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{49 \sqrt{42} - 42 ({\sqrt{6}}^{1} {\sqrt{6}}^{1}) + 6 \sqrt{7} (7 \sqrt{7}) + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.2.4

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

$\frac{49 \sqrt{42} - 42 {\sqrt{6}}^{1 + 1} + 6 \sqrt{7} (7 \sqrt{7}) + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.2.5

Add $1$ and $1$ .

$\frac{49 \sqrt{42} - 42 {\sqrt{6}}^{2} + 6 \sqrt{7} (7 \sqrt{7}) + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.3

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 4.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{49 \sqrt{42} - 42 {(6^{\frac{1}{2}})}^{2} + 6 \sqrt{7} (7 \sqrt{7}) + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.3.2

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

$\frac{49 \sqrt{42} - 42 \cdot 6^{\frac{1}{2} \cdot 2} + 6 \sqrt{7} (7 \sqrt{7}) + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.3.3

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

$\frac{49 \sqrt{42} - 42 \cdot 6^{\frac{2}{2}} + 6 \sqrt{7} (7 \sqrt{7}) + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.3.4

Cancel the common factor of $2$ .

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Step 4.1.3.4.1

Cancel the common factor.

$\frac{49 \sqrt{42} - 42 \cdot 6^{\frac{2}{2}} + 6 \sqrt{7} (7 \sqrt{7}) + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.3.4.2

Rewrite the expression.

$\frac{49 \sqrt{42} - 42 \cdot 6^{1} + 6 \sqrt{7} (7 \sqrt{7}) + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.3.5

Evaluate the exponent.

$\frac{49 \sqrt{42} - 42 \cdot 6 + 6 \sqrt{7} (7 \sqrt{7}) + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.4

Multiply $- 42$ by $6$ .

$\frac{49 \sqrt{42} - 252 + 6 \sqrt{7} (7 \sqrt{7}) + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.5

Multiply $6 \sqrt{7} (7 \sqrt{7})$ .

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Step 4.1.5.1

Multiply $7$ by $6$ .

$\frac{49 \sqrt{42} - 252 + 42 \sqrt{7} \sqrt{7} + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.5.2

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

$\frac{49 \sqrt{42} - 252 + 42 ({\sqrt{7}}^{1} \sqrt{7}) + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.5.3

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

$\frac{49 \sqrt{42} - 252 + 42 ({\sqrt{7}}^{1} {\sqrt{7}}^{1}) + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.5.4

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

$\frac{49 \sqrt{42} - 252 + 42 {\sqrt{7}}^{1 + 1} + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.5.5

Add $1$ and $1$ .

$\frac{49 \sqrt{42} - 252 + 42 {\sqrt{7}}^{2} + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.6

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

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Step 4.1.6.1

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

$\frac{49 \sqrt{42} - 252 + 42 {(7^{\frac{1}{2}})}^{2} + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.6.2

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

$\frac{49 \sqrt{42} - 252 + 42 \cdot 7^{\frac{1}{2} \cdot 2} + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.6.3

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

$\frac{49 \sqrt{42} - 252 + 42 \cdot 7^{\frac{2}{2}} + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.6.4

Cancel the common factor of $2$ .

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Step 4.1.6.4.1

Cancel the common factor.

$\frac{49 \sqrt{42} - 252 + 42 \cdot 7^{\frac{2}{2}} + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.6.4.2

Rewrite the expression.

$\frac{49 \sqrt{42} - 252 + 42 \cdot 7^{1} + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.6.5

Evaluate the exponent.

$\frac{49 \sqrt{42} - 252 + 42 \cdot 7 + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.7

Multiply $42$ by $7$ .

$\frac{49 \sqrt{42} - 252 + 294 + 6 \sqrt{7} (- 6 \sqrt{6})}{127}$

Step 4.1.8

Multiply $6 \sqrt{7} (- 6 \sqrt{6})$ .

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Step 4.1.8.1

Multiply $- 6$ by $6$ .

$\frac{49 \sqrt{42} - 252 + 294 - 36 \sqrt{7} \sqrt{6}}{127}$

Step 4.1.8.2

Combine using the product rule for radicals.

$\frac{49 \sqrt{42} - 252 + 294 - 36 \sqrt{6 \cdot 7}}{127}$

Step 4.1.8.3

Multiply $6$ by $7$ .

$\frac{49 \sqrt{42} - 252 + 294 - 36 \sqrt{42}}{127}$

Step 4.2

Subtract $36 \sqrt{42}$ from $49 \sqrt{42}$ .

$\frac{- 252 + 294 + 13 \sqrt{42}}{127}$

Step 4.3

Add $- 252$ and $294$ .

$\frac{42 + 13 \sqrt{42}}{127}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{42 + 13 \sqrt{42}}{127}$

Decimal Form:

$0.99409156 \dots$