Basic Math Examples

$\sqrt{20} (3 - \sqrt{45})$

Step 1

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

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Step 1.1

Factor $4$ out of $20$ .

$\sqrt{4 (5)} (3 - \sqrt{45})$

Step 1.2

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

$\sqrt{2^{2} \cdot 5} (3 - \sqrt{45})$

Step 2

Pull terms out from under the radical.

$2 \sqrt{5} (3 - \sqrt{45})$

Step 3

Simplify each term.

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Step 3.1

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 3.1.1

Factor $9$ out of $45$ .

$2 \sqrt{5} (3 - \sqrt{9 (5)})$

Step 3.1.2

Rewrite $9$ as $3^{2}$ .

$2 \sqrt{5} (3 - \sqrt{3^{2} \cdot 5})$

Step 3.2

Pull terms out from under the radical.

$2 \sqrt{5} (3 - (3 \sqrt{5}))$

Step 3.3

Multiply $3$ by $- 1$ .

$2 \sqrt{5} (3 - 3 \sqrt{5})$

Step 4

Simplify by multiplying through.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $3$ by $2$ .

$6 \sqrt{5} + 2 \sqrt{5} (- 3 \sqrt{5})$

Step 5

Multiply $2 \sqrt{5} (- 3 \sqrt{5})$ .

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Step 5.1

Multiply $- 3$ by $2$ .

$6 \sqrt{5} - 6 \sqrt{5} \sqrt{5}$

Step 5.2

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

$6 \sqrt{5} - 6 ({\sqrt{5}}^{1} \sqrt{5})$

Step 5.3

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

$6 \sqrt{5} - 6 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})$

Step 5.4

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

$6 \sqrt{5} - 6 {\sqrt{5}}^{1 + 1}$

Step 5.5

Add $1$ and $1$ .

$6 \sqrt{5} - 6 {\sqrt{5}}^{2}$

Step 6

Simplify each term.

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.1.3

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

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

Step 6.1.4

Cancel the common factor of $2$ .

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Step 6.1.4.1

Cancel the common factor.

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

Step 6.1.4.2

Rewrite the expression.

$6 \sqrt{5} - 6 \cdot 5^{1}$

Step 6.1.5

Evaluate the exponent.

$6 \sqrt{5} - 6 \cdot 5$

Step 6.2

Multiply $- 6$ by $5$ .

$6 \sqrt{5} - 30$

Step 7

The result can be shown in multiple forms.

Exact Form:

$6 \sqrt{5} - 30$

Decimal Form:

$- 16.58359213 \dots$