Linear Algebra Examples

$\frac{\sqrt{- 2} \sqrt{- 8} + \sqrt[3]{- 125}}{\sqrt{25} - \sqrt{- 9}}$

Step 1

Pull out imaginary unit $i$ .

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Step 1.1

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{\sqrt{- 1 (2)} \sqrt{- 8} + \sqrt[3]{- 125}}{\sqrt{25} - \sqrt{- 9}}$

Step 1.2

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$\frac{\sqrt{- 1} \cdot \sqrt{2} \sqrt{- 8} + \sqrt[3]{- 125}}{\sqrt{25} - \sqrt{- 9}}$

Step 1.3

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

$\frac{i \cdot \sqrt{2} \sqrt{- 8} + \sqrt[3]{- 125}}{\sqrt{25} - \sqrt{- 9}}$

Step 1.4

Rewrite $- 8$ as $- 1 (8)$ .

$\frac{i \cdot \sqrt{2} \sqrt{- 1 (8)} + \sqrt[3]{- 125}}{\sqrt{25} - \sqrt{- 9}}$

Step 1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$\frac{i \cdot \sqrt{2} (\sqrt{- 1} \cdot \sqrt{8}) + \sqrt[3]{- 125}}{\sqrt{25} - \sqrt{- 9}}$

Step 1.6

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

$\frac{i \cdot \sqrt{2} (i \cdot \sqrt{8}) + \sqrt[3]{- 125}}{\sqrt{25} - \sqrt{- 9}}$

Step 1.7

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{i \cdot \sqrt{2} (i \cdot \sqrt{8}) + \sqrt[3]{- 125}}{\sqrt{25} - \sqrt{- 1 (9)}}$

Step 1.8

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$\frac{i \cdot \sqrt{2} (i \cdot \sqrt{8}) + \sqrt[3]{- 125}}{\sqrt{25} - (\sqrt{- 1} \cdot \sqrt{9})}$

Step 1.9

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

$\frac{i \cdot \sqrt{2} (i \cdot \sqrt{8}) + \sqrt[3]{- 125}}{\sqrt{25} - (i \cdot \sqrt{9})}$

Step 2

Multiply $\frac{i \cdot \sqrt{2} i \cdot \sqrt{8} + \sqrt[3]{- 125}}{\sqrt{25} - i \cdot \sqrt{9}}$ by $\frac{\sqrt{25} + i \cdot \sqrt{9}}{\sqrt{25} + i \cdot \sqrt{9}}$ .

$\frac{i \cdot \sqrt{2} i \cdot \sqrt{8} + \sqrt[3]{- 125}}{\sqrt{25} - i \cdot \sqrt{9}} \cdot \frac{\sqrt{25} + i \cdot \sqrt{9}}{\sqrt{25} + i \cdot \sqrt{9}}$

Step 3

Combine fractions.

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Step 3.1

Multiply $\frac{i \cdot \sqrt{2} i \cdot \sqrt{8} + \sqrt[3]{- 125}}{\sqrt{25} - i \cdot \sqrt{9}}$ by $\frac{\sqrt{25} + i \cdot \sqrt{9}}{\sqrt{25} + i \cdot \sqrt{9}}$ .

$\frac{(i \cdot \sqrt{2} i \cdot \sqrt{8} + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{(\sqrt{25} - i \cdot \sqrt{9}) (\sqrt{25} + i \cdot \sqrt{9})}$

Step 3.2

Expand the denominator using the FOIL method.

$\frac{(i \cdot \sqrt{2} i \cdot \sqrt{8} + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{{\sqrt{25}}^{2} + \sqrt{225} i - i \sqrt{225} - i^{2} {\sqrt{9}}^{2}}$

Step 3.3

Simplify.

$\frac{(i \cdot \sqrt{2} i \cdot \sqrt{8} + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4

Simplify each term.

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Step 4.1

Multiply $i \sqrt{2} i$ .

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Step 4.1.1

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

$\frac{(i^{1} i \sqrt{2} \cdot \sqrt{8} + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4.1.2

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

$\frac{(i^{1} i^{1} \sqrt{2} \cdot \sqrt{8} + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4.1.3

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

$\frac{(i^{1 + 1} \sqrt{2} \cdot \sqrt{8} + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4.1.4

Add $1$ and $1$ .

$\frac{(i^{2} \sqrt{2} \cdot \sqrt{8} + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4.2

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

$\frac{(- 1 \sqrt{2} \cdot \sqrt{8} + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4.3

Rewrite $- 1 \sqrt{2}$ as $- \sqrt{2}$ .

$\frac{(- \sqrt{2} \cdot \sqrt{8} + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 4.4.1

Factor $4$ out of $8$ .

$\frac{(- \sqrt{2} \cdot \sqrt{4 (2)} + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4.4.2

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

$\frac{(- \sqrt{2} \cdot \sqrt{2^{2} \cdot 2} + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4.5

Pull terms out from under the radical.

$\frac{(- \sqrt{2} \cdot (2 \sqrt{2}) + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4.6

Multiply $- \sqrt{2} (2 \sqrt{2})$ .

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Step 4.6.1

Multiply $2$ by $- 1$ .

$\frac{(- 2 \sqrt{2} \sqrt{2} + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4.6.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{(- 2 ({\sqrt{2}}^{1} \sqrt{2}) + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4.6.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{(- 2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4.6.4

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

$\frac{(- 2 {\sqrt{2}}^{1 + 1} + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4.6.5

Add $1$ and $1$ .

$\frac{(- 2 {\sqrt{2}}^{2} + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4.7

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

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Step 4.7.1

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

$\frac{(- 2 {(2^{\frac{1}{2}})}^{2} + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4.7.2

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

$\frac{(- 2 \cdot 2^{\frac{1}{2} \cdot 2} + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4.7.3

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

$\frac{(- 2 \cdot 2^{\frac{2}{2}} + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4.7.4

Cancel the common factor of $2$ .

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Step 4.7.4.1

Cancel the common factor.

$\frac{(- 2 \cdot 2^{\frac{2}{2}} + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4.7.4.2

Rewrite the expression.

$\frac{(- 2 \cdot 2^{1} + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4.7.5

Evaluate the exponent.

$\frac{(- 2 \cdot 2 + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4.8

Multiply $- 2$ by $2$ .

$\frac{(- 4 + \sqrt[3]{- 125}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4.9

Rewrite $- 125$ as ${(- 5)}^{3}$ .

$\frac{(- 4 + \sqrt[3]{{(- 5)}^{3}}) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 4.10

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

$\frac{(- 4 - 5) (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 5

Subtract $5$ from $- 4$ .

$\frac{- 9 (\sqrt{25} + i \cdot \sqrt{9})}{34}$

Step 6

Simplify each term.

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Step 6.1

Rewrite $25$ as $5^{2}$ .

$\frac{- 9 (\sqrt{5^{2}} + i \cdot \sqrt{9})}{34}$

Step 6.2

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

$\frac{- 9 (5 + i \cdot \sqrt{9})}{34}$

Step 6.3

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

$\frac{- 9 (5 + i \cdot \sqrt{3^{2}})}{34}$

Step 6.4

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

$\frac{- 9 (5 + i \cdot 3)}{34}$

Step 6.5

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

$\frac{- 9 (5 + 3 i)}{34}$

Step 7

Simplify terms.

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Step 7.1

Apply the distributive property.

$\frac{- 9 \cdot 5 - 9 (3 i)}{34}$

Step 7.2

Multiply.

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Step 7.2.1

Multiply $- 9$ by $5$ .

$\frac{- 45 - 9 (3 i)}{34}$

Step 7.2.2

Multiply $3$ by $- 9$ .

$\frac{- 45 - 27 i}{34}$

Step 7.3

Rewrite $- 45$ as $- 1 (45)$ .

$\frac{- 1 (45) - 27 i}{34}$

Step 7.4

Factor $- 1$ out of $- 27 i$ .

$\frac{- 1 (45) - (27 i)}{34}$

Step 7.5

Factor $- 1$ out of $- 1 (45) - (27 i)$ .

$\frac{- 1 (45 + 27 i)}{34}$

Step 7.6

Move the negative in front of the fraction.

$- \frac{45 + 27 i}{34}$