Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

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Step 1.7.1

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

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

Step 1.7.2

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

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

Step 1.7.3

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

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

Step 1.7.4

Add $1$ and $1$ .

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

Step 1.7.5

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

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

Step 1.7.6

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

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

Step 1.7.7

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

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

Step 1.7.8

Add $1$ and $1$ .

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

Step 1.8

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

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

Step 1.9

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

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Step 1.9.1

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

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

Step 1.9.2

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

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

Step 1.9.3

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

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

Step 1.9.4

Cancel the common factor of $2$ .

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Step 1.9.4.1

Cancel the common factor.

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

Step 1.9.4.2

Rewrite the expression.

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

Step 1.9.5

Evaluate the exponent.

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

Step 1.10

Multiply $- 1$ by $10$ .

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

Step 1.11

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

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

Step 1.12

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

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

Step 1.13

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

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

Step 1.14

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

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

Step 1.15

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

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

Step 1.16

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

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

Step 1.17

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

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Step 1.17.1

Factor $9$ out of $27$ .

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

Step 1.17.2

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

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

Step 1.18

Pull terms out from under the radical.

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

Step 1.19

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

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

Step 1.20

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

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Step 1.20.1

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

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

Step 1.20.2

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

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

Step 1.20.3

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

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

Step 1.20.4

Add $1$ and $1$ .

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

Step 1.20.5

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

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

Step 1.20.6

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

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

Step 1.20.7

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

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

Step 1.20.8

Add $1$ and $1$ .

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

Step 1.21

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

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

Step 1.22

Multiply $3$ by $- 1$ .

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

Step 1.23

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

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Step 1.23.1

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

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

Step 1.23.2

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

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

Step 1.23.3

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

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

Step 1.23.4

Cancel the common factor of $2$ .

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Step 1.23.4.1

Cancel the common factor.

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

Step 1.23.4.2

Rewrite the expression.

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

Step 1.23.5

Evaluate the exponent.

$- 10 - 3 \cdot 3$

Step 1.24

Multiply $- 3$ by $3$ .

$- 10 - 9$

Step 2

Subtract $9$ from $- 10$ .

$- 19$