Evaluate (2 square root of 5+ square root of 10)/(4 square root of 5- square root of 10)

$\frac{2 \sqrt{5} + \sqrt{10}}{4 \sqrt{5} - \sqrt{10}}$

Step 1

Multiply $\frac{2 \sqrt{5} + \sqrt{10}}{4 \sqrt{5} - \sqrt{10}}$ by $\frac{4 \sqrt{5} + \sqrt{10}}{4 \sqrt{5} + \sqrt{10}}$ .

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

Step 2

Combine fractions.

Multiply $\frac{2 \sqrt{5} + \sqrt{10}}{4 \sqrt{5} - \sqrt{10}}$ by $\frac{4 \sqrt{5} + \sqrt{10}}{4 \sqrt{5} + \sqrt{10}}$ .

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

Expand the denominator using the FOIL method.

$\frac{(2 \sqrt{5} + \sqrt{10}) (4 \sqrt{5} + \sqrt{10})}{16 {\sqrt{5}}^{2} + 4 \sqrt{50} - 4 \sqrt{50} - {\sqrt{10}}^{2}}$

Simplify.

$\frac{(2 \sqrt{5} + \sqrt{10}) (4 \sqrt{5} + \sqrt{10})}{70}$

Step 3

Expand $(2 \sqrt{5} + \sqrt{10}) (4 \sqrt{5} + \sqrt{10})$ using the FOIL Method.

Apply the distributive property.

$\frac{2 \sqrt{5} (4 \sqrt{5} + \sqrt{10}) + \sqrt{10} (4 \sqrt{5} + \sqrt{10})}{70}$

Apply the distributive property.

$\frac{2 \sqrt{5} (4 \sqrt{5}) + 2 \sqrt{5} \sqrt{10} + \sqrt{10} (4 \sqrt{5} + \sqrt{10})}{70}$

Apply the distributive property.

$\frac{2 \sqrt{5} (4 \sqrt{5}) + 2 \sqrt{5} \sqrt{10} + \sqrt{10} (4 \sqrt{5}) + \sqrt{10} \sqrt{10}}{70}$

Step 4

Simplify and combine like terms.

Simplify each term.

Multiply $2 \sqrt{5} (4 \sqrt{5})$ .

Multiply $4$ by $2$ .

$\frac{8 \sqrt{5} \sqrt{5} + 2 \sqrt{5} \sqrt{10} + \sqrt{10} (4 \sqrt{5}) + \sqrt{10} \sqrt{10}}{70}$

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

$\frac{8 ({\sqrt{5}}^{1} \sqrt{5}) + 2 \sqrt{5} \sqrt{10} + \sqrt{10} (4 \sqrt{5}) + \sqrt{10} \sqrt{10}}{70}$

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

$\frac{8 ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) + 2 \sqrt{5} \sqrt{10} + \sqrt{10} (4 \sqrt{5}) + \sqrt{10} \sqrt{10}}{70}$

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

$\frac{8 {\sqrt{5}}^{1 + 1} + 2 \sqrt{5} \sqrt{10} + \sqrt{10} (4 \sqrt{5}) + \sqrt{10} \sqrt{10}}{70}$

Add $1$ and $1$ .

$\frac{8 {\sqrt{5}}^{2} + 2 \sqrt{5} \sqrt{10} + \sqrt{10} (4 \sqrt{5}) + \sqrt{10} \sqrt{10}}{70}$

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

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

$\frac{8 {(5^{\frac{1}{2}})}^{2} + 2 \sqrt{5} \sqrt{10} + \sqrt{10} (4 \sqrt{5}) + \sqrt{10} \sqrt{10}}{70}$

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

$\frac{8 \cdot 5^{\frac{1}{2} \cdot 2} + 2 \sqrt{5} \sqrt{10} + \sqrt{10} (4 \sqrt{5}) + \sqrt{10} \sqrt{10}}{70}$

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

$\frac{8 \cdot 5^{\frac{2}{2}} + 2 \sqrt{5} \sqrt{10} + \sqrt{10} (4 \sqrt{5}) + \sqrt{10} \sqrt{10}}{70}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{8 \cdot 5^{\frac{2}{2}} + 2 \sqrt{5} \sqrt{10} + \sqrt{10} (4 \sqrt{5}) + \sqrt{10} \sqrt{10}}{70}$

Rewrite the expression.

$\frac{8 \cdot 5^{1} + 2 \sqrt{5} \sqrt{10} + \sqrt{10} (4 \sqrt{5}) + \sqrt{10} \sqrt{10}}{70}$

Evaluate the exponent.

$\frac{8 \cdot 5 + 2 \sqrt{5} \sqrt{10} + \sqrt{10} (4 \sqrt{5}) + \sqrt{10} \sqrt{10}}{70}$

Multiply $8$ by $5$ .

$\frac{40 + 2 \sqrt{5} \sqrt{10} + \sqrt{10} (4 \sqrt{5}) + \sqrt{10} \sqrt{10}}{70}$

Multiply $2 \sqrt{5} \sqrt{10}$ .

Combine using the product rule for radicals.

$\frac{40 + 2 \sqrt{10 \cdot 5} + \sqrt{10} (4 \sqrt{5}) + \sqrt{10} \sqrt{10}}{70}$

Multiply $10$ by $5$ .

$\frac{40 + 2 \sqrt{50} + \sqrt{10} (4 \sqrt{5}) + \sqrt{10} \sqrt{10}}{70}$

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

Factor $25$ out of $50$ .

$\frac{40 + 2 \sqrt{25 (2)} + \sqrt{10} (4 \sqrt{5}) + \sqrt{10} \sqrt{10}}{70}$

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

$\frac{40 + 2 \sqrt{5^{2} \cdot 2} + \sqrt{10} (4 \sqrt{5}) + \sqrt{10} \sqrt{10}}{70}$

Pull terms out from under the radical.

$\frac{40 + 2 (5 \sqrt{2}) + \sqrt{10} (4 \sqrt{5}) + \sqrt{10} \sqrt{10}}{70}$

Multiply $5$ by $2$ .

$\frac{40 + 10 \sqrt{2} + \sqrt{10} (4 \sqrt{5}) + \sqrt{10} \sqrt{10}}{70}$

Multiply $\sqrt{10} (4 \sqrt{5})$ .

Combine using the product rule for radicals.

$\frac{40 + 10 \sqrt{2} + 4 \sqrt{10 \cdot 5} + \sqrt{10} \sqrt{10}}{70}$

Multiply $10$ by $5$ .

$\frac{40 + 10 \sqrt{2} + 4 \sqrt{50} + \sqrt{10} \sqrt{10}}{70}$

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

Factor $25$ out of $50$ .

$\frac{40 + 10 \sqrt{2} + 4 \sqrt{25 (2)} + \sqrt{10} \sqrt{10}}{70}$

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

$\frac{40 + 10 \sqrt{2} + 4 \sqrt{5^{2} \cdot 2} + \sqrt{10} \sqrt{10}}{70}$

Pull terms out from under the radical.

$\frac{40 + 10 \sqrt{2} + 4 (5 \sqrt{2}) + \sqrt{10} \sqrt{10}}{70}$

Multiply $5$ by $4$ .

$\frac{40 + 10 \sqrt{2} + 20 \sqrt{2} + \sqrt{10} \sqrt{10}}{70}$

Combine using the product rule for radicals.

$\frac{40 + 10 \sqrt{2} + 20 \sqrt{2} + \sqrt{10 \cdot 10}}{70}$

Multiply $10$ by $10$ .

$\frac{40 + 10 \sqrt{2} + 20 \sqrt{2} + \sqrt{100}}{70}$

Rewrite $100$ as $10^{2}$ .

$\frac{40 + 10 \sqrt{2} + 20 \sqrt{2} + \sqrt{10^{2}}}{70}$

Pull terms out from under the radical, assuming positive real numbers.

$\frac{40 + 10 \sqrt{2} + 20 \sqrt{2} + 10}{70}$

Add $40$ and $10$ .

$\frac{50 + 10 \sqrt{2} + 20 \sqrt{2}}{70}$

Add $10 \sqrt{2}$ and $20 \sqrt{2}$ .

$\frac{50 + 30 \sqrt{2}}{70}$

Step 5

Cancel the common factor of $50 + 30 \sqrt{2}$ and $70$ .

Factor $10$ out of $50$ .

$\frac{10 (5) + 30 \sqrt{2}}{70}$

Factor $10$ out of $30 \sqrt{2}$ .

$\frac{10 (5) + 10 (3 \sqrt{2})}{70}$

Factor $10$ out of $10 (5) + 10 (3 \sqrt{2})$ .

$\frac{10 (5 + 3 \sqrt{2})}{70}$

Cancel the common factors.

Factor $10$ out of $70$ .

$\frac{10 (5 + 3 \sqrt{2})}{10 \cdot 7}$

Cancel the common factor.

$\frac{10 (5 + 3 \sqrt{2})}{10 \cdot 7}$

Rewrite the expression.

$\frac{5 + 3 \sqrt{2}}{7}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{5 + 3 \sqrt{2}}{7}$

Decimal Form:

$1.32037724 \dots$

Evaluate (2 square root of 5+ square root of 10)/(4 square root of 5- square root of 10)

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