Basic Math Examples

$(\sqrt{5 - \sqrt{2}}) (\sqrt{5 + \sqrt{2}})$

Step 1

Combine using the product rule for radicals.

$\sqrt{(5 - \sqrt{2}) (5 + \sqrt{2})}$

Step 2

Expand $(5 - \sqrt{2}) (5 + \sqrt{2})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\sqrt{5 (5 + \sqrt{2}) - \sqrt{2} (5 + \sqrt{2})}$

Step 2.2

Apply the distributive property.

$\sqrt{5 \cdot 5 + 5 \sqrt{2} - \sqrt{2} (5 + \sqrt{2})}$

Step 2.3

Apply the distributive property.

$\sqrt{5 \cdot 5 + 5 \sqrt{2} - \sqrt{2} \cdot 5 - \sqrt{2} \sqrt{2}}$

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 $5$ by $5$ .

$\sqrt{25 + 5 \sqrt{2} - \sqrt{2} \cdot 5 - \sqrt{2} \sqrt{2}}$

Step 3.1.2

Multiply $5$ by $- 1$ .

$\sqrt{25 + 5 \sqrt{2} - 5 \sqrt{2} - \sqrt{2} \sqrt{2}}$

Step 3.1.3

Multiply $- \sqrt{2} \sqrt{2}$ .

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Step 3.1.3.1

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt{25 + 5 \sqrt{2} - 5 \sqrt{2} - ({\sqrt{2}}^{1} \sqrt{2})}$

Step 3.1.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt{25 + 5 \sqrt{2} - 5 \sqrt{2} - ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 3.1.3.3

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

$\sqrt{25 + 5 \sqrt{2} - 5 \sqrt{2} - {\sqrt{2}}^{1 + 1}}$

Step 3.1.3.4

Add $1$ and $1$ .

$\sqrt{25 + 5 \sqrt{2} - 5 \sqrt{2} - {\sqrt{2}}^{2}}$

Step 3.1.4

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

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Step 3.1.4.1

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

$\sqrt{25 + 5 \sqrt{2} - 5 \sqrt{2} - {(2^{\frac{1}{2}})}^{2}}$

Step 3.1.4.2

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

$\sqrt{25 + 5 \sqrt{2} - 5 \sqrt{2} - 2^{\frac{1}{2} \cdot 2}}$

Step 3.1.4.3

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

$\sqrt{25 + 5 \sqrt{2} - 5 \sqrt{2} - 2^{\frac{2}{2}}}$

Step 3.1.4.4

Cancel the common factor of $2$ .

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Step 3.1.4.4.1

Cancel the common factor.

$\sqrt{25 + 5 \sqrt{2} - 5 \sqrt{2} - 2^{\frac{2}{2}}}$

Step 3.1.4.4.2

Rewrite the expression.

$\sqrt{25 + 5 \sqrt{2} - 5 \sqrt{2} - 2^{1}}$

Step 3.1.4.5

Evaluate the exponent.

$\sqrt{25 + 5 \sqrt{2} - 5 \sqrt{2} - 1 \cdot 2}$

Step 3.1.5

Multiply $- 1$ by $2$ .

$\sqrt{25 + 5 \sqrt{2} - 5 \sqrt{2} - 2}$

Step 3.2

Subtract $2$ from $25$ .

$\sqrt{23 + 5 \sqrt{2} - 5 \sqrt{2}}$

Step 3.3

Subtract $5 \sqrt{2}$ from $5 \sqrt{2}$ .

$\sqrt{23 + 0}$

Step 3.4

Add $23$ and $0$ .

$\sqrt{23}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\sqrt{23}$

Decimal Form:

$4.79583152 \dots$