Basic Math Examples

$\sqrt{5} (3 \sqrt{5} - \frac{1}{2}) + \sqrt{20}$

Step 1

Simplify each term.

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Step 1.1

To write $3 \sqrt{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{5} (3 \sqrt{5} \cdot \frac{2}{2} - \frac{1}{2}) + \sqrt{20}$

Step 1.2

Combine $3 \sqrt{5}$ and $\frac{2}{2}$ .

$\sqrt{5} (\frac{3 \sqrt{5} \cdot 2}{2} - \frac{1}{2}) + \sqrt{20}$

Step 1.3

Combine the numerators over the common denominator.

$\sqrt{5} \frac{3 \sqrt{5} \cdot 2 - 1}{2} + \sqrt{20}$

Step 1.4

Multiply $2$ by $3$ .

$\sqrt{5} \frac{6 \sqrt{5} - 1}{2} + \sqrt{20}$

Step 1.5

Combine $\sqrt{5}$ and $\frac{6 \sqrt{5} - 1}{2}$ .

$\frac{\sqrt{5} (6 \sqrt{5} - 1)}{2} + \sqrt{20}$

Step 1.6

Apply the distributive property.

$\frac{\sqrt{5} (6 \sqrt{5}) + \sqrt{5} \cdot - 1}{2} + \sqrt{20}$

Step 1.7

Multiply $\sqrt{5} (6 \sqrt{5})$ .

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Step 1.7.1

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

$\frac{6 ({\sqrt{5}}^{1} \sqrt{5}) + \sqrt{5} \cdot - 1}{2} + \sqrt{20}$

Step 1.7.2

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

$\frac{6 ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) + \sqrt{5} \cdot - 1}{2} + \sqrt{20}$

Step 1.7.3

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

$\frac{6 {\sqrt{5}}^{1 + 1} + \sqrt{5} \cdot - 1}{2} + \sqrt{20}$

Step 1.7.4

Add $1$ and $1$ .

$\frac{6 {\sqrt{5}}^{2} + \sqrt{5} \cdot - 1}{2} + \sqrt{20}$

Step 1.8

Move $- 1$ to the left of $\sqrt{5}$ .

$\frac{6 {\sqrt{5}}^{2} - 1 \cdot \sqrt{5}}{2} + \sqrt{20}$

Step 1.9

Simplify each term.

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Step 1.9.1

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

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Step 1.9.1.1

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

$\frac{6 {(5^{\frac{1}{2}})}^{2} - 1 \cdot \sqrt{5}}{2} + \sqrt{20}$

Step 1.9.1.2

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

$\frac{6 \cdot 5^{\frac{1}{2} \cdot 2} - 1 \cdot \sqrt{5}}{2} + \sqrt{20}$

Step 1.9.1.3

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

$\frac{6 \cdot 5^{\frac{2}{2}} - 1 \cdot \sqrt{5}}{2} + \sqrt{20}$

Step 1.9.1.4

Cancel the common factor of $2$ .

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Step 1.9.1.4.1

Cancel the common factor.

$\frac{6 \cdot 5^{\frac{2}{2}} - 1 \cdot \sqrt{5}}{2} + \sqrt{20}$

Step 1.9.1.4.2

Rewrite the expression.

$\frac{6 \cdot 5^{1} - 1 \cdot \sqrt{5}}{2} + \sqrt{20}$

Step 1.9.1.5

Evaluate the exponent.

$\frac{6 \cdot 5 - 1 \cdot \sqrt{5}}{2} + \sqrt{20}$

Step 1.9.2

Multiply $6$ by $5$ .

$\frac{30 - 1 \cdot \sqrt{5}}{2} + \sqrt{20}$

Step 1.9.3

Rewrite $- 1 \sqrt{5}$ as $- \sqrt{5}$ .

$\frac{30 - \sqrt{5}}{2} + \sqrt{20}$

Step 1.10

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

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Step 1.10.1

Factor $4$ out of $20$ .

$\frac{30 - \sqrt{5}}{2} + \sqrt{4 (5)}$

Step 1.10.2

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

$\frac{30 - \sqrt{5}}{2} + \sqrt{2^{2} \cdot 5}$

Step 1.11

Pull terms out from under the radical.

$\frac{30 - \sqrt{5}}{2} + 2 \sqrt{5}$

Step 2

To write $2 \sqrt{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{30 - \sqrt{5}}{2} + 2 \sqrt{5} \cdot \frac{2}{2}$

Step 3

Combine fractions.

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Step 3.1

Combine $2 \sqrt{5}$ and $\frac{2}{2}$ .

$\frac{30 - \sqrt{5}}{2} + \frac{2 \sqrt{5} \cdot 2}{2}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{30 - \sqrt{5} + 2 \sqrt{5} \cdot 2}{2}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $2$ by $2$ .

$\frac{30 - \sqrt{5} + 4 \sqrt{5}}{2}$

Step 4.2

Add $- \sqrt{5}$ and $4 \sqrt{5}$ .

$\frac{30 + 3 \sqrt{5}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{30 + 3 \sqrt{5}}{2}$

Decimal Form:

$18.35410196 \dots$