Simplify square root of -10 square root of -10+ square root of -3 square root of -27

$\sqrt{- 10} \sqrt{- 10} + \sqrt{- 3} \sqrt{- 27}$

Step 1

Simplify each term.

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

$\sqrt{- 1 (10)} \sqrt{- 10} + \sqrt{- 3} \sqrt{- 27}$

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

$\sqrt{- 1} \cdot \sqrt{10} \sqrt{- 10} + \sqrt{- 3} \sqrt{- 27}$

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

$i \cdot \sqrt{10} \sqrt{- 10} + \sqrt{- 3} \sqrt{- 27}$

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

$i \cdot \sqrt{10} \sqrt{- 1 (10)} + \sqrt{- 3} \sqrt{- 27}$

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

$i \cdot \sqrt{10} (\sqrt{- 1} \cdot \sqrt{10}) + \sqrt{- 3} \sqrt{- 27}$

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

$i \cdot \sqrt{10} (i \cdot \sqrt{10}) + \sqrt{- 3} \sqrt{- 27}$

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

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

$i^{1} i \sqrt{10} \sqrt{10} + \sqrt{- 3} \sqrt{- 27}$

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

$i^{1} i^{1} \sqrt{10} \sqrt{10} + \sqrt{- 3} \sqrt{- 27}$

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

$i^{1 + 1} \sqrt{10} \sqrt{10} + \sqrt{- 3} \sqrt{- 27}$

Add $1$ and $1$ .

$i^{2} \sqrt{10} \sqrt{10} + \sqrt{- 3} \sqrt{- 27}$

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

$i^{2} ({\sqrt{10}}^{1} \sqrt{10}) + \sqrt{- 3} \sqrt{- 27}$

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

$i^{2} ({\sqrt{10}}^{1} {\sqrt{10}}^{1}) + \sqrt{- 3} \sqrt{- 27}$

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

$i^{2} {\sqrt{10}}^{1 + 1} + \sqrt{- 3} \sqrt{- 27}$

Add $1$ and $1$ .

$i^{2} {\sqrt{10}}^{2} + \sqrt{- 3} \sqrt{- 27}$

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

$- 1 {\sqrt{10}}^{2} + \sqrt{- 3} \sqrt{- 27}$

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

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

$- 1 {(10^{\frac{1}{2}})}^{2} + \sqrt{- 3} \sqrt{- 27}$

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

$- 1 \cdot 10^{\frac{1}{2} \cdot 2} + \sqrt{- 3} \sqrt{- 27}$

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

$- 1 \cdot 10^{\frac{2}{2}} + \sqrt{- 3} \sqrt{- 27}$

Cancel the common factor of $2$ .

Cancel the common factor.

$- 1 \cdot 10^{\frac{2}{2}} + \sqrt{- 3} \sqrt{- 27}$

Rewrite the expression.

$- 1 \cdot 10^{1} + \sqrt{- 3} \sqrt{- 27}$

Evaluate the exponent.

$- 1 \cdot 10 + \sqrt{- 3} \sqrt{- 27}$

Multiply $- 1$ by $10$ .

$- 10 + \sqrt{- 3} \sqrt{- 27}$

Rewrite $- 3$ as $- 1 (3)$ .

$- 10 + \sqrt{- 1 (3)} \sqrt{- 27}$

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$- 10 + \sqrt{- 1} \cdot \sqrt{3} \sqrt{- 27}$

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

$- 10 + i \cdot \sqrt{3} \sqrt{- 27}$

Rewrite $- 27$ as $- 1 (27)$ .

$- 10 + i \cdot \sqrt{3} \sqrt{- 1 (27)}$

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$- 10 + i \cdot \sqrt{3} (\sqrt{- 1} \cdot \sqrt{27})$

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

$- 10 + i \cdot \sqrt{3} (i \cdot \sqrt{27})$

Rewrite $27$ as $3^{2} \cdot 3$ .

Factor $9$ out of $27$ .

$- 10 + i \sqrt{3} (i \cdot \sqrt{9 (3)})$

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

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

Pull terms out from under the radical.

$- 10 + i \sqrt{3} (i \cdot (3 \sqrt{3}))$

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

$- 10 + i \sqrt{3} (3 \cdot i \sqrt{3})$

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

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

$- 10 + \sqrt{3} (3 (i^{1} i) \sqrt{3})$

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

$- 10 + \sqrt{3} (3 (i^{1} i^{1}) \sqrt{3})$

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

$- 10 + \sqrt{3} (3 i^{1 + 1} \sqrt{3})$

Add $1$ and $1$ .

$- 10 + \sqrt{3} (3 i^{2} \sqrt{3})$

Raise $\sqrt{3}$ to the power of $1$ .

$- 10 + 3 i^{2} ({\sqrt{3}}^{1} \sqrt{3})$

Raise $\sqrt{3}$ to the power of $1$ .

$- 10 + 3 i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

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

$- 10 + 3 i^{2} {\sqrt{3}}^{1 + 1}$

Add $1$ and $1$ .

$- 10 + 3 i^{2} {\sqrt{3}}^{2}$

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

$- 10 + 3 \cdot - 1 {\sqrt{3}}^{2}$

Multiply $3$ by $- 1$ .

$- 10 - 3 {\sqrt{3}}^{2}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$- 10 - 3 {(3^{\frac{1}{2}})}^{2}$

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

$- 10 - 3 \cdot 3^{\frac{1}{2} \cdot 2}$

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

$- 10 - 3 \cdot 3^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$- 10 - 3 \cdot 3^{\frac{2}{2}}$

Rewrite the expression.

$- 10 - 3 \cdot 3^{1}$

Evaluate the exponent.

$- 10 - 3 \cdot 3$

Multiply $- 3$ by $3$ .

$- 10 - 9$

Step 2

Subtract $9$ from $- 10$ .

$- 19$

Simplify square root of -10 square root of -10+ square root of -3 square root of -27

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