Simplify the Radical Expression (4 square root of 20+6 square root of 30-10 square root of 40)/(2 square root of 10)

$\frac{4 \sqrt{20} + 6 \sqrt{30} - 10 \sqrt{40}}{2 \sqrt{10}}$

Step 1

Cancel the common factor of $4 \sqrt{20} + 6 \sqrt{30} - 10 \sqrt{40}$ and $2$ .

Factor $2$ out of $4 \sqrt{20}$ .

$\frac{2 (2 \sqrt{20}) + 6 \sqrt{30} - 10 \sqrt{40}}{2 \sqrt{10}}$

Factor $2$ out of $6 \sqrt{30}$ .

$\frac{2 (2 \sqrt{20}) + 2 (3 \sqrt{30}) - 10 \sqrt{40}}{2 \sqrt{10}}$

Factor $2$ out of $2 (2 \sqrt{20}) + 2 (3 \sqrt{30})$ .

$\frac{2 (2 \sqrt{20} + 3 \sqrt{30}) - 10 \sqrt{40}}{2 \sqrt{10}}$

Factor $2$ out of $- 10 \sqrt{40}$ .

$\frac{2 (2 \sqrt{20} + 3 \sqrt{30}) + 2 (- 5 \sqrt{40})}{2 \sqrt{10}}$

Factor $2$ out of $2 (2 \sqrt{20} + 3 \sqrt{30}) + 2 (- 5 \sqrt{40})$ .

$\frac{2 (2 \sqrt{20} + 3 \sqrt{30} - 5 \sqrt{40})}{2 \sqrt{10}}$

Cancel the common factors.

Factor $2$ out of $2 \sqrt{10}$ .

$\frac{2 (2 \sqrt{20} + 3 \sqrt{30} - 5 \sqrt{40})}{2 (\sqrt{10})}$

Cancel the common factor.

$\frac{2 (2 \sqrt{20} + 3 \sqrt{30} - 5 \sqrt{40})}{2 \sqrt{10}}$

Rewrite the expression.

$\frac{2 \sqrt{20} + 3 \sqrt{30} - 5 \sqrt{40}}{\sqrt{10}}$

Step 2

Simplify the numerator.

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

Factor $4$ out of $20$ .

$\frac{2 \sqrt{4 (5)} + 3 \sqrt{30} - 5 \sqrt{40}}{\sqrt{10}}$

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

$\frac{2 \sqrt{2^{2} \cdot 5} + 3 \sqrt{30} - 5 \sqrt{40}}{\sqrt{10}}$

Pull terms out from under the radical.

$\frac{2 (| 2 | \sqrt{5}) + 3 \sqrt{30} - 5 \sqrt{40}}{\sqrt{10}}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 (2 \sqrt{5}) + 3 \sqrt{30} - 5 \sqrt{40}}{\sqrt{10}}$

Multiply $2$ by $2$ .

$\frac{4 \sqrt{5} + 3 \sqrt{30} - 5 \sqrt{40}}{\sqrt{10}}$

Rewrite $40$ as $2^{2} \cdot 10$ .

Factor $4$ out of $40$ .

$\frac{4 \sqrt{5} + 3 \sqrt{30} - 5 \sqrt{4 (10)}}{\sqrt{10}}$

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

$\frac{4 \sqrt{5} + 3 \sqrt{30} - 5 \sqrt{2^{2} \cdot 10}}{\sqrt{10}}$

Pull terms out from under the radical.

$\frac{4 \sqrt{5} + 3 \sqrt{30} - 5 (| 2 | \sqrt{10})}{\sqrt{10}}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{4 \sqrt{5} + 3 \sqrt{30} - 5 (2 \sqrt{10})}{\sqrt{10}}$

Multiply $2$ by $- 5$ .

$\frac{4 \sqrt{5} + 3 \sqrt{30} - 10 \sqrt{10}}{\sqrt{10}}$

Step 3

Multiply $\frac{4 \sqrt{5} + 3 \sqrt{30} - 10 \sqrt{10}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{4 \sqrt{5} + 3 \sqrt{30} - 10 \sqrt{10}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{4 \sqrt{5} + 3 \sqrt{30} - 10 \sqrt{10}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{(4 \sqrt{5} + 3 \sqrt{30} - 10 \sqrt{10}) \sqrt{10}}{\sqrt{10} \sqrt{10}}$

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

$\frac{(4 \sqrt{5} + 3 \sqrt{30} - 10 \sqrt{10}) \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}}$

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

$\frac{(4 \sqrt{5} + 3 \sqrt{30} - 10 \sqrt{10}) \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}}$

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

$\frac{(4 \sqrt{5} + 3 \sqrt{30} - 10 \sqrt{10}) \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{(4 \sqrt{5} + 3 \sqrt{30} - 10 \sqrt{10}) \sqrt{10}}{{\sqrt{10}}^{2}}$

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

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

$\frac{(4 \sqrt{5} + 3 \sqrt{30} - 10 \sqrt{10}) \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

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

$\frac{(4 \sqrt{5} + 3 \sqrt{30} - 10 \sqrt{10}) \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

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

$\frac{(4 \sqrt{5} + 3 \sqrt{30} - 10 \sqrt{10}) \sqrt{10}}{10^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{(4 \sqrt{5} + 3 \sqrt{30} - 10 \sqrt{10}) \sqrt{10}}{10^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{(4 \sqrt{5} + 3 \sqrt{30} - 10 \sqrt{10}) \sqrt{10}}{10^{1}}$

Evaluate the exponent.

$\frac{(4 \sqrt{5} + 3 \sqrt{30} - 10 \sqrt{10}) \sqrt{10}}{10}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{(4 \sqrt{5} + 3 \sqrt{30} - 10 \sqrt{10}) \sqrt{10}}{10}$

Decimal Form:

$- 1.97542045 \dots$

Simplify the Radical Expression (4 square root of 20+6 square root of 30-10 square root of 40)/(2 square root of 10)

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