Pre-Algebra Examples

$(\sqrt{7} - \sqrt{13}) (\sqrt{7} + \sqrt{13})$

Step 1

Expand $(\sqrt{7} - \sqrt{13}) (\sqrt{7} + \sqrt{13})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$\sqrt{7} (\sqrt{7} + \sqrt{13}) - \sqrt{13} (\sqrt{7} + \sqrt{13})$

Step 1.2

Apply the distributive property.

$\sqrt{7} \sqrt{7} + \sqrt{7} \sqrt{13} - \sqrt{13} (\sqrt{7} + \sqrt{13})$

Step 1.3

Apply the distributive property.

$\sqrt{7} \sqrt{7} + \sqrt{7} \sqrt{13} - \sqrt{13} \sqrt{7} - \sqrt{13} \sqrt{13}$

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

Combine using the product rule for radicals.

$\sqrt{7 \cdot 7} + \sqrt{7} \sqrt{13} - \sqrt{13} \sqrt{7} - \sqrt{13} \sqrt{13}$

Step 2.1.2

Multiply $7$ by $7$ .

$\sqrt{49} + \sqrt{7} \sqrt{13} - \sqrt{13} \sqrt{7} - \sqrt{13} \sqrt{13}$

Step 2.1.3

Rewrite $49$ as $7^{2}$ .

$\sqrt{7^{2}} + \sqrt{7} \sqrt{13} - \sqrt{13} \sqrt{7} - \sqrt{13} \sqrt{13}$

Step 2.1.4

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

$7 + \sqrt{7} \sqrt{13} - \sqrt{13} \sqrt{7} - \sqrt{13} \sqrt{13}$

Step 2.1.5

Combine using the product rule for radicals.

$7 + \sqrt{7 \cdot 13} - \sqrt{13} \sqrt{7} - \sqrt{13} \sqrt{13}$

Step 2.1.6

Multiply $7$ by $13$ .

$7 + \sqrt{91} - \sqrt{13} \sqrt{7} - \sqrt{13} \sqrt{13}$

Step 2.1.7

Multiply $- \sqrt{13} \sqrt{7}$ .

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Step 2.1.7.1

Combine using the product rule for radicals.

$7 + \sqrt{91} - \sqrt{7 \cdot 13} - \sqrt{13} \sqrt{13}$

Step 2.1.7.2

Multiply $7$ by $13$ .

$7 + \sqrt{91} - \sqrt{91} - \sqrt{13} \sqrt{13}$

Step 2.1.8

Multiply $- \sqrt{13} \sqrt{13}$ .

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Step 2.1.8.1

Raise $\sqrt{13}$ to the power of $1$ .

$7 + \sqrt{91} - \sqrt{91} - ({\sqrt{13}}^{1} \sqrt{13})$

Step 2.1.8.2

Raise $\sqrt{13}$ to the power of $1$ .

$7 + \sqrt{91} - \sqrt{91} - ({\sqrt{13}}^{1} {\sqrt{13}}^{1})$

Step 2.1.8.3

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

$7 + \sqrt{91} - \sqrt{91} - {\sqrt{13}}^{1 + 1}$

Step 2.1.8.4

Add $1$ and $1$ .

$7 + \sqrt{91} - \sqrt{91} - {\sqrt{13}}^{2}$

Step 2.1.9

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 2.1.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$7 + \sqrt{91} - \sqrt{91} - {(13^{\frac{1}{2}})}^{2}$

Step 2.1.9.2

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

$7 + \sqrt{91} - \sqrt{91} - 13^{\frac{1}{2} \cdot 2}$

Step 2.1.9.3

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

$7 + \sqrt{91} - \sqrt{91} - 13^{\frac{2}{2}}$

Step 2.1.9.4

Cancel the common factor of $2$ .

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Step 2.1.9.4.1

Cancel the common factor.

$7 + \sqrt{91} - \sqrt{91} - 13^{\frac{2}{2}}$

Step 2.1.9.4.2

Rewrite the expression.

$7 + \sqrt{91} - \sqrt{91} - 13^{1}$

Step 2.1.9.5

Evaluate the exponent.

$7 + \sqrt{91} - \sqrt{91} - 1 \cdot 13$

Step 2.1.10

Multiply $- 1$ by $13$ .

$7 + \sqrt{91} - \sqrt{91} - 13$

Step 2.2

Subtract $13$ from $7$ .

$- 6 + \sqrt{91} - \sqrt{91}$

Step 2.3

Subtract $\sqrt{91}$ from $\sqrt{91}$ .

$- 6 + 0$

Step 2.4

Add $- 6$ and $0$ .

$- 6$