Algebra Examples

$\sqrt{14 - 2 \sqrt{13}} \cdot \sqrt{14 + 2 \sqrt{13}}$

Step 1

Combine using the product rule for radicals.

$\sqrt{(14 - 2 \sqrt{13}) \cdot (14 + 2 \sqrt{13})}$

Step 2

Expand $(14 - 2 \sqrt{13}) (14 + 2 \sqrt{13})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\sqrt{14 (14 + 2 \sqrt{13}) - 2 \sqrt{13} (14 + 2 \sqrt{13})}$

Step 2.2

Apply the distributive property.

$\sqrt{14 \cdot 14 + 14 (2 \sqrt{13}) - 2 \sqrt{13} (14 + 2 \sqrt{13})}$

Step 2.3

Apply the distributive property.

$\sqrt{14 \cdot 14 + 14 (2 \sqrt{13}) - 2 \sqrt{13} \cdot 14 - 2 \sqrt{13} (2 \sqrt{13})}$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $14$ by $14$ .

$\sqrt{196 + 14 (2 \sqrt{13}) - 2 \sqrt{13} \cdot 14 - 2 \sqrt{13} (2 \sqrt{13})}$

Step 3.1.2

Multiply $2$ by $14$ .

$\sqrt{196 + 28 \sqrt{13} - 2 \sqrt{13} \cdot 14 - 2 \sqrt{13} (2 \sqrt{13})}$

Step 3.1.3

Multiply $14$ by $- 2$ .

$\sqrt{196 + 28 \sqrt{13} - 28 \sqrt{13} - 2 \sqrt{13} (2 \sqrt{13})}$

Step 3.1.4

Multiply $- 2 \sqrt{13} (2 \sqrt{13})$ .

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Step 3.1.4.1

Multiply $2$ by $- 2$ .

$\sqrt{196 + 28 \sqrt{13} - 28 \sqrt{13} - 4 \sqrt{13} \sqrt{13}}$

Step 3.1.4.2

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

$\sqrt{196 + 28 \sqrt{13} - 28 \sqrt{13} - 4 ({\sqrt{13}}^{1} \sqrt{13})}$

Step 3.1.4.3

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

$\sqrt{196 + 28 \sqrt{13} - 28 \sqrt{13} - 4 ({\sqrt{13}}^{1} {\sqrt{13}}^{1})}$

Step 3.1.4.4

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

$\sqrt{196 + 28 \sqrt{13} - 28 \sqrt{13} - 4 {\sqrt{13}}^{1 + 1}}$

Step 3.1.4.5

Add $1$ and $1$ .

$\sqrt{196 + 28 \sqrt{13} - 28 \sqrt{13} - 4 {\sqrt{13}}^{2}}$

Step 3.1.5

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

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Step 3.1.5.1

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

$\sqrt{196 + 28 \sqrt{13} - 28 \sqrt{13} - 4 {(13^{\frac{1}{2}})}^{2}}$

Step 3.1.5.2

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

$\sqrt{196 + 28 \sqrt{13} - 28 \sqrt{13} - 4 \cdot 13^{\frac{1}{2} \cdot 2}}$

Step 3.1.5.3

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

$\sqrt{196 + 28 \sqrt{13} - 28 \sqrt{13} - 4 \cdot 13^{\frac{2}{2}}}$

Step 3.1.5.4

Cancel the common factor of $2$ .

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Step 3.1.5.4.1

Cancel the common factor.

$\sqrt{196 + 28 \sqrt{13} - 28 \sqrt{13} - 4 \cdot 13^{\frac{2}{2}}}$

Step 3.1.5.4.2

Rewrite the expression.

$\sqrt{196 + 28 \sqrt{13} - 28 \sqrt{13} - 4 \cdot 13^{1}}$

Step 3.1.5.5

Evaluate the exponent.

$\sqrt{196 + 28 \sqrt{13} - 28 \sqrt{13} - 4 \cdot 13}$

Step 3.1.6

Multiply $- 4$ by $13$ .

$\sqrt{196 + 28 \sqrt{13} - 28 \sqrt{13} - 52}$

Step 3.2

Subtract $52$ from $196$ .

$\sqrt{144 + 28 \sqrt{13} - 28 \sqrt{13}}$

Step 3.3

Subtract $28 \sqrt{13}$ from $28 \sqrt{13}$ .

$\sqrt{144 + 0}$

Step 3.4

Add $144$ and $0$ .

$\sqrt{144}$

Step 4

Rewrite $144$ as $12^{2}$ .

$\sqrt{12^{2}}$

Step 5

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

$12$