Basic Math Examples

$\frac{\sqrt{11} - \sqrt{7}}{\sqrt{11} + \sqrt{7}} = a - b \sqrt{77}$

Step 1

Rewrite the equation as $a - b \sqrt{77} = \frac{\sqrt{11} - \sqrt{7}}{\sqrt{11} + \sqrt{7}}$ .

$a - b \sqrt{77} = \frac{\sqrt{11} - \sqrt{7}}{\sqrt{11} + \sqrt{7}}$

Step 2

Simplify $\frac{\sqrt{11} - \sqrt{7}}{\sqrt{11} + \sqrt{7}}$ .

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Step 2.1

Multiply $\frac{\sqrt{11} - \sqrt{7}}{\sqrt{11} + \sqrt{7}}$ by $\frac{\sqrt{11} - \sqrt{7}}{\sqrt{11} - \sqrt{7}}$ .

$a - b \sqrt{77} = \frac{\sqrt{11} - \sqrt{7}}{\sqrt{11} + \sqrt{7}} \cdot \frac{\sqrt{11} - \sqrt{7}}{\sqrt{11} - \sqrt{7}}$

Step 2.2

Combine fractions.

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Step 2.2.1

Multiply $\frac{\sqrt{11} - \sqrt{7}}{\sqrt{11} + \sqrt{7}}$ by $\frac{\sqrt{11} - \sqrt{7}}{\sqrt{11} - \sqrt{7}}$ .

$a - b \sqrt{77} = \frac{(\sqrt{11} - \sqrt{7}) (\sqrt{11} - \sqrt{7})}{(\sqrt{11} + \sqrt{7}) (\sqrt{11} - \sqrt{7})}$

Step 2.2.2

Expand the denominator using the FOIL method.

$a - b \sqrt{77} = \frac{(\sqrt{11} - \sqrt{7}) (\sqrt{11} - \sqrt{7})}{{\sqrt{11}}^{2} - \sqrt{77} + \sqrt{77} - {\sqrt{7}}^{2}}$

Step 2.2.3

Simplify.

$a - b \sqrt{77} = \frac{(\sqrt{11} - \sqrt{7}) (\sqrt{11} - \sqrt{7})}{4}$

Step 2.3

Simplify the numerator.

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Step 2.3.1

Raise $\sqrt{11} - \sqrt{7}$ to the power of $1$ .

$a - b \sqrt{77} = \frac{{(\sqrt{11} - \sqrt{7})}^{1} (\sqrt{11} - \sqrt{7})}{4}$

Step 2.3.2

Raise $\sqrt{11} - \sqrt{7}$ to the power of $1$ .

$a - b \sqrt{77} = \frac{{(\sqrt{11} - \sqrt{7})}^{1} {(\sqrt{11} - \sqrt{7})}^{1}}{4}$

Step 2.3.3

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

$a - b \sqrt{77} = \frac{{(\sqrt{11} - \sqrt{7})}^{1 + 1}}{4}$

Step 2.3.4

Add $1$ and $1$ .

$a - b \sqrt{77} = \frac{{(\sqrt{11} - \sqrt{7})}^{2}}{4}$

Step 2.4

Rewrite ${(\sqrt{11} - \sqrt{7})}^{2}$ as $(\sqrt{11} - \sqrt{7}) (\sqrt{11} - \sqrt{7})$ .

$a - b \sqrt{77} = \frac{(\sqrt{11} - \sqrt{7}) (\sqrt{11} - \sqrt{7})}{4}$

Step 2.5

Expand $(\sqrt{11} - \sqrt{7}) (\sqrt{11} - \sqrt{7})$ using the FOIL Method.

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Step 2.5.1

Apply the distributive property.

$a - b \sqrt{77} = \frac{\sqrt{11} (\sqrt{11} - \sqrt{7}) - \sqrt{7} (\sqrt{11} - \sqrt{7})}{4}$

Step 2.5.2

Apply the distributive property.

$a - b \sqrt{77} = \frac{\sqrt{11} \sqrt{11} + \sqrt{11} (- \sqrt{7}) - \sqrt{7} (\sqrt{11} - \sqrt{7})}{4}$

Step 2.5.3

Apply the distributive property.

$a - b \sqrt{77} = \frac{\sqrt{11} \sqrt{11} + \sqrt{11} (- \sqrt{7}) - \sqrt{7} \sqrt{11} - \sqrt{7} (- \sqrt{7})}{4}$

Step 2.6

Simplify and combine like terms.

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Step 2.6.1

Simplify each term.

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Step 2.6.1.1

Combine using the product rule for radicals.

$a - b \sqrt{77} = \frac{\sqrt{11 \cdot 11} + \sqrt{11} (- \sqrt{7}) - \sqrt{7} \sqrt{11} - \sqrt{7} (- \sqrt{7})}{4}$

Step 2.6.1.2

Multiply $11$ by $11$ .

$a - b \sqrt{77} = \frac{\sqrt{121} + \sqrt{11} (- \sqrt{7}) - \sqrt{7} \sqrt{11} - \sqrt{7} (- \sqrt{7})}{4}$

Step 2.6.1.3

Rewrite $121$ as $11^{2}$ .

$a - b \sqrt{77} = \frac{\sqrt{11^{2}} + \sqrt{11} (- \sqrt{7}) - \sqrt{7} \sqrt{11} - \sqrt{7} (- \sqrt{7})}{4}$

Step 2.6.1.4

Pull terms out from under the radical, assuming positive real numbers.

$a - b \sqrt{77} = \frac{11 + \sqrt{11} (- \sqrt{7}) - \sqrt{7} \sqrt{11} - \sqrt{7} (- \sqrt{7})}{4}$

Step 2.6.1.5

Multiply $\sqrt{11} (- \sqrt{7})$ .

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Step 2.6.1.5.1

Combine using the product rule for radicals.

$a - b \sqrt{77} = \frac{11 - \sqrt{11 \cdot 7} - \sqrt{7} \sqrt{11} - \sqrt{7} (- \sqrt{7})}{4}$

Step 2.6.1.5.2

Multiply $11$ by $7$ .

$a - b \sqrt{77} = \frac{11 - \sqrt{77} - \sqrt{7} \sqrt{11} - \sqrt{7} (- \sqrt{7})}{4}$

Step 2.6.1.6

Multiply $- \sqrt{7} \sqrt{11}$ .

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Step 2.6.1.6.1

Combine using the product rule for radicals.

$a - b \sqrt{77} = \frac{11 - \sqrt{77} - \sqrt{11 \cdot 7} - \sqrt{7} (- \sqrt{7})}{4}$

Step 2.6.1.6.2

Multiply $11$ by $7$ .

$a - b \sqrt{77} = \frac{11 - \sqrt{77} - \sqrt{77} - \sqrt{7} (- \sqrt{7})}{4}$

Step 2.6.1.7

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

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Step 2.6.1.7.1

Multiply $- 1$ by $- 1$ .

$a - b \sqrt{77} = \frac{11 - \sqrt{77} - \sqrt{77} + 1 \sqrt{7} \sqrt{7}}{4}$

Step 2.6.1.7.2

Multiply $\sqrt{7}$ by $1$ .

$a - b \sqrt{77} = \frac{11 - \sqrt{77} - \sqrt{77} + \sqrt{7} \sqrt{7}}{4}$

Step 2.6.1.7.3

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

$a - b \sqrt{77} = \frac{11 - \sqrt{77} - \sqrt{77} + {\sqrt{7}}^{1} \sqrt{7}}{4}$

Step 2.6.1.7.4

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

$a - b \sqrt{77} = \frac{11 - \sqrt{77} - \sqrt{77} + {\sqrt{7}}^{1} {\sqrt{7}}^{1}}{4}$

Step 2.6.1.7.5

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

$a - b \sqrt{77} = \frac{11 - \sqrt{77} - \sqrt{77} + {\sqrt{7}}^{1 + 1}}{4}$

Step 2.6.1.7.6

Add $1$ and $1$ .

$a - b \sqrt{77} = \frac{11 - \sqrt{77} - \sqrt{77} + {\sqrt{7}}^{2}}{4}$

Step 2.6.1.8

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

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Step 2.6.1.8.1

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

$a - b \sqrt{77} = \frac{11 - \sqrt{77} - \sqrt{77} + {(7^{\frac{1}{2}})}^{2}}{4}$

Step 2.6.1.8.2

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

$a - b \sqrt{77} = \frac{11 - \sqrt{77} - \sqrt{77} + 7^{\frac{1}{2} \cdot 2}}{4}$

Step 2.6.1.8.3

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

$a - b \sqrt{77} = \frac{11 - \sqrt{77} - \sqrt{77} + 7^{\frac{2}{2}}}{4}$

Step 2.6.1.8.4

Cancel the common factor of $2$ .

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Step 2.6.1.8.4.1

Cancel the common factor.

$a - b \sqrt{77} = \frac{11 - \sqrt{77} - \sqrt{77} + 7^{\frac{2}{2}}}{4}$

Step 2.6.1.8.4.2

Rewrite the expression.

$a - b \sqrt{77} = \frac{11 - \sqrt{77} - \sqrt{77} + 7^{1}}{4}$

Step 2.6.1.8.5

Evaluate the exponent.

$a - b \sqrt{77} = \frac{11 - \sqrt{77} - \sqrt{77} + 7}{4}$

Step 2.6.2

Add $11$ and $7$ .

$a - b \sqrt{77} = \frac{18 - \sqrt{77} - \sqrt{77}}{4}$

Step 2.6.3

Subtract $\sqrt{77}$ from $- \sqrt{77}$ .

$a - b \sqrt{77} = \frac{18 - 2 \sqrt{77}}{4}$

Step 2.7

Cancel the common factor of $18 - 2 \sqrt{77}$ and $4$ .

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Step 2.7.1

Factor $2$ out of $18$ .

$a - b \sqrt{77} = \frac{2 (9) - 2 \sqrt{77}}{4}$

Step 2.7.2

Factor $2$ out of $- 2 \sqrt{77}$ .

$a - b \sqrt{77} = \frac{2 (9) + 2 (- \sqrt{77})}{4}$

Step 2.7.3

Factor $2$ out of $2 (9) + 2 (- \sqrt{77})$ .

$a - b \sqrt{77} = \frac{2 (9 - \sqrt{77})}{4}$

Step 2.7.4

Cancel the common factors.

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Step 2.7.4.1

Factor $2$ out of $4$ .

$a - b \sqrt{77} = \frac{2 (9 - \sqrt{77})}{2 \cdot 2}$

Step 2.7.4.2

Cancel the common factor.

$a - b \sqrt{77} = \frac{2 (9 - \sqrt{77})}{2 \cdot 2}$

Step 2.7.4.3

Rewrite the expression.

$a - b \sqrt{77} = \frac{9 - \sqrt{77}}{2}$

Step 3

Add $b \sqrt{77}$ to both sides of the equation.

$a = \frac{9 - \sqrt{77}}{2} + b \sqrt{77}$