Simplify ( square root of -5+2 square root of 20)(- square root of -5)

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

Step 1

Rewrite $- 5$ as $- 1 (5)$ .

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

Step 2

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

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

Step 3

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

$(i \cdot \sqrt{5} + 2 \sqrt{20}) (- \sqrt{- 5})$

Step 4

Rewrite $- 5$ as $- 1 (5)$ .

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

Step 5

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

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

Step 6

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

$(i \cdot \sqrt{5} + 2 \sqrt{20}) (- (i \cdot \sqrt{5}))$

Step 7

Simplify each term.

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

Factor $4$ out of $20$ .

$(i \sqrt{5} + 2 \sqrt{4 (5)}) (- (i \cdot \sqrt{5}))$

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $2$ .

$(i \sqrt{5} + 4 \sqrt{5}) (- (i \cdot \sqrt{5}))$

Step 8

Apply the distributive property.

$i \sqrt{5} (- i \sqrt{5}) + 4 \sqrt{5} (- i \sqrt{5})$

Step 9

Multiply $i \sqrt{5} (- i \sqrt{5})$ .

Raise $i$ to the power of $1$ .

$\sqrt{5} (- (i^{1} i) \sqrt{5}) + 4 \sqrt{5} (- i \sqrt{5})$

Raise $i$ to the power of $1$ .

$\sqrt{5} (- (i^{1} i^{1}) \sqrt{5}) + 4 \sqrt{5} (- i \sqrt{5})$

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

$\sqrt{5} (- i^{1 + 1} \sqrt{5}) + 4 \sqrt{5} (- i \sqrt{5})$

Add $1$ and $1$ .

$\sqrt{5} (- i^{2} \sqrt{5}) + 4 \sqrt{5} (- i \sqrt{5})$

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

$- i^{2} ({\sqrt{5}}^{1} \sqrt{5}) + 4 \sqrt{5} (- i \sqrt{5})$

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

$- i^{2} ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) + 4 \sqrt{5} (- i \sqrt{5})$

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

$- i^{2} {\sqrt{5}}^{1 + 1} + 4 \sqrt{5} (- i \sqrt{5})$

Add $1$ and $1$ .

$- i^{2} {\sqrt{5}}^{2} + 4 \sqrt{5} (- i \sqrt{5})$

Step 10

Multiply $4 \sqrt{5} (- i \sqrt{5})$ .

Multiply $- 1$ by $4$ .

$- i^{2} {\sqrt{5}}^{2} - 4 \sqrt{5} (i \sqrt{5})$

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

$- i^{2} {\sqrt{5}}^{2} - 4 ({\sqrt{5}}^{1} \sqrt{5}) i$

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

$- i^{2} {\sqrt{5}}^{2} - 4 ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) i$

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

$- i^{2} {\sqrt{5}}^{2} - 4 {\sqrt{5}}^{1 + 1} i$

Add $1$ and $1$ .

$- i^{2} {\sqrt{5}}^{2} - 4 {\sqrt{5}}^{2} i$

Step 11

Simplify each term.

Rewrite $i^{2}$ as $- 1$ .

$- - 1 {\sqrt{5}}^{2} - 4 {\sqrt{5}}^{2} i$

Multiply $- 1$ by $- 1$ .

$1 {\sqrt{5}}^{2} - 4 {\sqrt{5}}^{2} i$

Multiply ${\sqrt{5}}^{2}$ by $1$ .

${\sqrt{5}}^{2} - 4 {\sqrt{5}}^{2} i$

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

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

${(5^{\frac{1}{2}})}^{2} - 4 {\sqrt{5}}^{2} i$

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

$5^{\frac{1}{2} \cdot 2} - 4 {\sqrt{5}}^{2} i$

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

$5^{\frac{2}{2}} - 4 {\sqrt{5}}^{2} i$

Cancel the common factor of $2$ .

Cancel the common factor.

$5^{\frac{2}{2}} - 4 {\sqrt{5}}^{2} i$

Rewrite the expression.

$5^{1} - 4 {\sqrt{5}}^{2} i$

Evaluate the exponent.

$5 - 4 {\sqrt{5}}^{2} i$

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

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

$5 - 4 {(5^{\frac{1}{2}})}^{2} i$

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

$5 - 4 \cdot 5^{\frac{1}{2} \cdot 2} i$

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

$5 - 4 \cdot 5^{\frac{2}{2}} i$

Cancel the common factor of $2$ .

Cancel the common factor.

$5 - 4 \cdot 5^{\frac{2}{2}} i$

Rewrite the expression.

$5 - 4 \cdot 5^{1} i$

Evaluate the exponent.

$5 - 4 \cdot 5 i$

Multiply $- 4$ by $5$ .

$5 - 20 i$

Simplify ( square root of -5+2 square root of 20)(- square root of -5)

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