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

$3$ .

$\frac{21 i}{2}$