Basic Math Examples

$4 (3 + 2 \sqrt{5}) - \sqrt{10} (\sqrt{2} + \sqrt{10})$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$4 \cdot 3 + 4 (2 \sqrt{5}) - \sqrt{10} (\sqrt{2} + \sqrt{10})$

Step 1.2

Multiply $4$ by $3$ .

$12 + 4 (2 \sqrt{5}) - \sqrt{10} (\sqrt{2} + \sqrt{10})$

Step 1.3

Multiply $2$ by $4$ .

$12 + 8 \sqrt{5} - \sqrt{10} (\sqrt{2} + \sqrt{10})$

Step 1.4

Apply the distributive property.

$12 + 8 \sqrt{5} - \sqrt{10} \sqrt{2} - \sqrt{10} \sqrt{10}$

Step 1.5

Multiply $- \sqrt{10} \sqrt{2}$ .

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Step 1.5.1

Combine using the product rule for radicals.

$12 + 8 \sqrt{5} - \sqrt{2 \cdot 10} - \sqrt{10} \sqrt{10}$

Step 1.5.2

Multiply $2$ by $10$ .

$12 + 8 \sqrt{5} - \sqrt{20} - \sqrt{10} \sqrt{10}$

Step 1.6

Multiply $- \sqrt{10} \sqrt{10}$ .

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Step 1.6.1

Raise $\sqrt{10}$ to the power of $1$ .

$12 + 8 \sqrt{5} - \sqrt{20} - ({\sqrt{10}}^{1} \sqrt{10})$

Step 1.6.2

Raise $\sqrt{10}$ to the power of $1$ .

$12 + 8 \sqrt{5} - \sqrt{20} - ({\sqrt{10}}^{1} {\sqrt{10}}^{1})$

Step 1.6.3

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

$12 + 8 \sqrt{5} - \sqrt{20} - {\sqrt{10}}^{1 + 1}$

Step 1.6.4

Add $1$ and $1$ .

$12 + 8 \sqrt{5} - \sqrt{20} - {\sqrt{10}}^{2}$

Step 1.7

Simplify each term.

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Step 1.7.1

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 1.7.1.1

Factor $4$ out of $20$ .

$12 + 8 \sqrt{5} - \sqrt{4 (5)} - {\sqrt{10}}^{2}$

Step 1.7.1.2

Rewrite $4$ as $2^{2}$ .

$12 + 8 \sqrt{5} - \sqrt{2^{2} \cdot 5} - {\sqrt{10}}^{2}$

Step 1.7.2

Pull terms out from under the radical.

$12 + 8 \sqrt{5} - (2 \sqrt{5}) - {\sqrt{10}}^{2}$

Step 1.7.3

Multiply $2$ by $- 1$ .

$12 + 8 \sqrt{5} - 2 \sqrt{5} - {\sqrt{10}}^{2}$

Step 1.7.4

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 1.7.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$12 + 8 \sqrt{5} - 2 \sqrt{5} - {(10^{\frac{1}{2}})}^{2}$

Step 1.7.4.2

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

$12 + 8 \sqrt{5} - 2 \sqrt{5} - 10^{\frac{1}{2} \cdot 2}$

Step 1.7.4.3

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

$12 + 8 \sqrt{5} - 2 \sqrt{5} - 10^{\frac{2}{2}}$

Step 1.7.4.4

Cancel the common factor of $2$ .

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Step 1.7.4.4.1

Cancel the common factor.

$12 + 8 \sqrt{5} - 2 \sqrt{5} - 10^{\frac{2}{2}}$

Step 1.7.4.4.2

Rewrite the expression.

$12 + 8 \sqrt{5} - 2 \sqrt{5} - 10^{1}$

Step 1.7.4.5

Evaluate the exponent.

$12 + 8 \sqrt{5} - 2 \sqrt{5} - 1 \cdot 10$

Step 1.7.5

Multiply $- 1$ by $10$ .

$12 + 8 \sqrt{5} - 2 \sqrt{5} - 10$

Step 2

Simplify by adding terms.

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Step 2.1

Subtract $10$ from $12$ .

$2 + 8 \sqrt{5} - 2 \sqrt{5}$

Step 2.2

Subtract $2 \sqrt{5}$ from $8 \sqrt{5}$ .

$2 + 6 \sqrt{5}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$2 + 6 \sqrt{5}$

Decimal Form:

$15.41640786 \dots$