Linear Algebra Examples

$\frac{\sqrt{- 36} \sqrt{- 49}}{\sqrt{- 16}}$

Step 1

Pull out imaginary unit $i$ .

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Step 1.1

Rewrite $- 36$ as $- 1 (36)$ .

$\frac{\sqrt{- 1 (36)} \sqrt{- 49}}{\sqrt{- 16}}$

Step 1.2

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$\frac{\sqrt{- 1} \cdot \sqrt{36} \sqrt{- 49}}{\sqrt{- 16}}$

Step 1.3

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

$\frac{i \cdot \sqrt{36} \sqrt{- 49}}{\sqrt{- 16}}$

Step 1.4

Rewrite $- 49$ as $- 1 (49)$ .

$\frac{i \cdot \sqrt{36} \sqrt{- 1 (49)}}{\sqrt{- 16}}$

Step 1.5

Rewrite $\sqrt{- 1 (49)}$ as $\sqrt{- 1} \cdot \sqrt{49}$ .

$\frac{i \cdot \sqrt{36} (\sqrt{- 1} \cdot \sqrt{49})}{\sqrt{- 16}}$

Step 1.6

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

$\frac{i \cdot \sqrt{36} (i \cdot \sqrt{49})}{\sqrt{- 16}}$

Step 1.7

Rewrite $- 16$ as $- 1 (16)$ .

$\frac{i \cdot \sqrt{36} (i \cdot \sqrt{49})}{\sqrt{- 1 (16)}}$

Step 1.8

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$\frac{i \cdot \sqrt{36} (i \cdot \sqrt{49})}{\sqrt{- 1} \cdot \sqrt{16}}$

Step 1.9

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

$\frac{i \cdot \sqrt{36} (i \cdot \sqrt{49})}{i \cdot \sqrt{16}}$

Step 2

Cancel the common factor of $i$ .

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Step 2.1

Cancel the common factor.

$\frac{i \cdot \sqrt{36} (i \cdot \sqrt{49})}{i \cdot \sqrt{16}}$

Step 2.2

Rewrite the expression.

$\frac{\sqrt{36} (i \cdot \sqrt{49})}{\sqrt{16}}$

Step 3

Combine $\sqrt{36}$ and $\sqrt{16}$ into a single radical.

$\sqrt{\frac{36}{16}} i \cdot \sqrt{49}$

Step 4

Cancel the common factor of $36$ and $16$ .

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Step 4.1

Factor $4$ out of $36$ .

$\sqrt{\frac{4 (9)}{16}} i \cdot \sqrt{49}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $4$ out of $16$ .

$\sqrt{\frac{4 \cdot 9}{4 \cdot 4}} i \cdot \sqrt{49}$

Step 4.2.2

Cancel the common factor.

$\sqrt{\frac{4 \cdot 9}{4 \cdot 4}} i \cdot \sqrt{49}$

Step 4.2.3

Rewrite the expression.

$\sqrt{\frac{9}{4}} i \cdot \sqrt{49}$

Step 5

Rewrite $\sqrt{\frac{9}{4}}$ as $\frac{\sqrt{9}}{\sqrt{4}}$ .

$\frac{\sqrt{9}}{\sqrt{4}} i \cdot \sqrt{49}$

Step 6

Simplify the numerator.

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Step 6.1

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

$\frac{\sqrt{3^{2}}}{\sqrt{4}} i \cdot \sqrt{49}$

Step 6.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{3}{\sqrt{4}} i \cdot \sqrt{49}$

Step 7

Simplify the denominator.

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Step 7.1

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

$\frac{3}{\sqrt{2^{2}}} i \cdot \sqrt{49}$

Step 7.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{3}{2} i \cdot \sqrt{49}$

Step 8

Combine fractions.

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Step 8.1

Combine $\frac{3}{2}$ and $i$ .

$\frac{3 i}{2} \cdot \sqrt{49}$

Step 8.2

Rewrite $49$ as $7^{2}$ .

$\frac{3 i}{2} \cdot \sqrt{7^{2}}$

Step 9

Pull terms out from under the radical, assuming positive real numbers.

$\frac{3 i}{2} \cdot 7$

Step 10

Multiply $\frac{3 i}{2} \cdot 7$ .

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Step 10.1

Combine $\frac{3 i}{2}$ and $7$ .

$\frac{3 i \cdot 7}{2}$

Step 10.2

Multiply $7$ by $3$ .

$\frac{21 i}{2}$