Basic Math Examples

$\sqrt{5} (\sqrt{10} + 3 \sqrt{5})$

Step 1

Apply the distributive property.

$\sqrt{5} \sqrt{10} + \sqrt{5} (3 \sqrt{5})$

Step 2

Combine using the product rule for radicals.

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

Step 3

Multiply $\sqrt{5} (3 \sqrt{5})$ .

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Step 3.1

Raise $\sqrt{5}$ to the power of $1$ .

$\sqrt{5 \cdot 10} + 3 ({\sqrt{5}}^{1} \sqrt{5})$

Step 3.2

Raise $\sqrt{5}$ to the power of $1$ .

$\sqrt{5 \cdot 10} + 3 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})$

Step 3.3

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

$\sqrt{5 \cdot 10} + 3 {\sqrt{5}}^{1 + 1}$

Step 3.4

Add $1$ and $1$ .

$\sqrt{5 \cdot 10} + 3 {\sqrt{5}}^{2}$

Step 4

Simplify each term.

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Step 4.1

Multiply $5$ by $10$ .

$\sqrt{50} + 3 {\sqrt{5}}^{2}$

Step 4.2

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 4.2.1

Factor $25$ out of $50$ .

$\sqrt{25 (2)} + 3 {\sqrt{5}}^{2}$

Step 4.2.2

Rewrite $25$ as $5^{2}$ .

$\sqrt{5^{2} \cdot 2} + 3 {\sqrt{5}}^{2}$

Step 4.3

Pull terms out from under the radical.

$5 \sqrt{2} + 3 {\sqrt{5}}^{2}$

Step 4.4

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$5 \sqrt{2} + 3 {(5^{\frac{1}{2}})}^{2}$

Step 4.4.2

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

$5 \sqrt{2} + 3 \cdot 5^{\frac{1}{2} \cdot 2}$

Step 4.4.3

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

$5 \sqrt{2} + 3 \cdot 5^{\frac{2}{2}}$

Step 4.4.4

Cancel the common factor of $2$ .

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Step 4.4.4.1

Cancel the common factor.

$5 \sqrt{2} + 3 \cdot 5^{\frac{2}{2}}$

Step 4.4.4.2

Rewrite the expression.

$5 \sqrt{2} + 3 \cdot 5^{1}$

Step 4.4.5

Evaluate the exponent.

$5 \sqrt{2} + 3 \cdot 5$

Step 4.5

Multiply $3$ by $5$ .

$5 \sqrt{2} + 15$

Step 5

The result can be shown in multiple forms.

Exact Form:

$5 \sqrt{2} + 15$

Decimal Form:

$22.07106781 \dots$