Simplify ( square root of -10)(6 square root of 90)

$(\sqrt{- 10}) (6 \sqrt{90})$

Step 1

Rewrite $- 10$ as $- 1 (10)$ .

$\sqrt{- 1 (10)} (6 \sqrt{90})$

Step 2

Rewrite $\sqrt{- 1 (10)}$ as $\sqrt{- 1} \cdot \sqrt{10}$ .

$\sqrt{- 1} \cdot \sqrt{10} (6 \sqrt{90})$

Step 3

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

$i \cdot \sqrt{10} (6 \sqrt{90})$

Step 4

Rewrite $90$ as $3^{2} \cdot 10$ .

Factor $9$ out of $90$ .

$i \sqrt{10} (6 \sqrt{9 (10)})$

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

$i \sqrt{10} (6 \sqrt{3^{2} \cdot 10})$

Step 5

Pull terms out from under the radical.

$i \sqrt{10} (6 (3 \sqrt{10}))$

Step 6

Multiply $3$ by $6$ .

$i \sqrt{10} (18 \sqrt{10})$

Step 7

Multiply $i \sqrt{10} (18 \sqrt{10})$ .

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

$i (18 ({\sqrt{10}}^{1} \sqrt{10}))$

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

$i (18 ({\sqrt{10}}^{1} {\sqrt{10}}^{1}))$

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

$i (18 {\sqrt{10}}^{1 + 1})$

Add $1$ and $1$ .

$i (18 {\sqrt{10}}^{2})$

Step 8

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

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

$i (18 {(10^{\frac{1}{2}})}^{2})$

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

$i (18 \cdot 10^{\frac{1}{2} \cdot 2})$

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

$i (18 \cdot 10^{\frac{2}{2}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$i (18 \cdot 10^{\frac{2}{2}})$

Rewrite the expression.

$i \cdot 18 \cdot 10^{1}$

Evaluate the exponent.

$i (18 \cdot 10)$

Step 9

Simplify the expression.

Multiply $18$ by $10$ .

$i \cdot 180$

Move $180$ to the left of $i$ .

$180 i$

Simplify ( square root of -10)(6 square root of 90)

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