Simplify square root of -5 square root of -45

$\sqrt{- 5} \sqrt{- 45}$

Step 1

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

$\sqrt{- 1 (5)} \sqrt{- 45}$

Step 2

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

$\sqrt{- 1} \cdot \sqrt{5} \sqrt{- 45}$

Step 3

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

$i \cdot \sqrt{5} \sqrt{- 45}$

Step 4

Rewrite $- 45$ as $- 1 (45)$ .

$i \cdot \sqrt{5} \sqrt{- 1 (45)}$

Step 5

Rewrite $\sqrt{- 1 (45)}$ as $\sqrt{- 1} \cdot \sqrt{45}$ .

$i \cdot \sqrt{5} (\sqrt{- 1} \cdot \sqrt{45})$

Step 6

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

$i \cdot \sqrt{5} (i \cdot \sqrt{45})$

Step 7

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

Factor $9$ out of $45$ .

$i \sqrt{5} (i \cdot \sqrt{9 (5)})$

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

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

Step 8

Pull terms out from under the radical.

$i \sqrt{5} (i \cdot (3 \sqrt{5}))$

Step 9

Move $3$ to the left of $i$ .

$i \sqrt{5} (3 \cdot i \sqrt{5})$

Step 10

Multiply $i \sqrt{5} (3 i \sqrt{5})$ .

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

$\sqrt{5} (3 (i^{1} i) \sqrt{5})$

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

$\sqrt{5} (3 (i^{1} i^{1}) \sqrt{5})$

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

$\sqrt{5} (3 i^{1 + 1} \sqrt{5})$

Add $1$ and $1$ .

$\sqrt{5} (3 i^{2} \sqrt{5})$

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

$3 i^{2} ({\sqrt{5}}^{1} \sqrt{5})$

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

$3 i^{2} ({\sqrt{5}}^{1} {\sqrt{5}}^{1})$

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

$3 i^{2} {\sqrt{5}}^{1 + 1}$

Add $1$ and $1$ .

$3 i^{2} {\sqrt{5}}^{2}$

Step 11

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

$3 \cdot - 1 {\sqrt{5}}^{2}$

Step 12

Multiply $3$ by $- 1$ .

$- 3 {\sqrt{5}}^{2}$

Step 13

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

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

$- 3 {(5^{\frac{1}{2}})}^{2}$

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

$- 3 \cdot 5^{\frac{1}{2} \cdot 2}$

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

$- 3 \cdot 5^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$- 3 \cdot 5^{\frac{2}{2}}$

Rewrite the expression.

$- 3 \cdot 5^{1}$

Evaluate the exponent.

$- 3 \cdot 5$

Step 14

Multiply $- 3$ by $5$ .

$- 15$

Simplify square root of -5 square root of -45

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