Algebra Examples

$\frac{\sqrt{- 49}}{(7 - 2 i) - (4 + 9 i)}$

Step 1

Multiply to rationalize the numerator.

$\frac{\sqrt{- 49} \sqrt{- 49}}{((7 - 2 i) - (4 + 9 i)) \sqrt{- 49}}$

Step 2

Simplify.

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Step 2.1

Raise $\sqrt{- 49}$ to the power of $1$ .

$\frac{{\sqrt{- 49}}^{1} \sqrt{- 49}}{((7 - 2 i) - (4 + 9 i)) \sqrt{- 49}}$

Step 2.2

Raise $\sqrt{- 49}$ to the power of $1$ .

$\frac{{\sqrt{- 49}}^{1} {\sqrt{- 49}}^{1}}{((7 - 2 i) - (4 + 9 i)) \sqrt{- 49}}$

Step 2.3

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

$\frac{{\sqrt{- 49}}^{1 + 1}}{((7 - 2 i) - (4 + 9 i)) \sqrt{- 49}}$

Step 2.4

Add $1$ and $1$ .

$\frac{{\sqrt{- 49}}^{2}}{((7 - 2 i) - (4 + 9 i)) \sqrt{- 49}}$

Step 2.5

Rewrite ${\sqrt{- 49}}^{2}$ as $- 49$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- 49}$ as ${(- 49)}^{\frac{1}{2}}$ .

$\frac{{({(- 49)}^{\frac{1}{2}})}^{2}}{((7 - 2 i) - (4 + 9 i)) \sqrt{- 49}}$

Step 2.5.2

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

$\frac{{(- 49)}^{\frac{1}{2} \cdot 2}}{((7 - 2 i) - (4 + 9 i)) \sqrt{- 49}}$

Step 2.5.3

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

$\frac{{(- 49)}^{\frac{2}{2}}}{((7 - 2 i) - (4 + 9 i)) \sqrt{- 49}}$

Step 2.5.4

Cancel the common factor of $2$ .

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Step 2.5.4.1

Cancel the common factor.

$\frac{{(- 49)}^{\frac{2}{2}}}{((7 - 2 i) - (4 + 9 i)) \sqrt{- 49}}$

Step 2.5.4.2

Rewrite the expression.

$\frac{{(- 49)}^{1}}{((7 - 2 i) - (4 + 9 i)) \sqrt{- 49}}$

Step 2.5.5

Evaluate the exponent.

$\frac{- 49}{((7 - 2 i) - (4 + 9 i)) \sqrt{- 49}}$

Step 2.6

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

$\frac{- 49}{((7 - 2 i) - (4 + 9 i)) \sqrt{- 1 (49)}}$

Step 2.7

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

$\frac{- 49}{((7 - 2 i) - (4 + 9 i)) (\sqrt{- 1} \cdot \sqrt{49})}$

Step 2.8

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

$\frac{- 49}{((7 - 2 i) - (4 + 9 i)) (i \cdot \sqrt{49})}$

Step 2.9

Simplify each term.

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Step 2.9.1

Apply the distributive property.

$\frac{- 49}{(7 - 2 i - 1 \cdot 4 - (9 i)) (i \cdot \sqrt{49})}$

Step 2.9.2

Multiply $- 1$ by $4$ .

$\frac{- 49}{(7 - 2 i - 4 - (9 i)) (i \cdot \sqrt{49})}$

Step 2.9.3

Multiply $9$ by $- 1$ .

$\frac{- 49}{(7 - 2 i - 4 - 9 i) (i \cdot \sqrt{49})}$

Step 2.10

Subtract $4$ from $7$ .

$\frac{- 49}{(3 - 2 i - 9 i) (i \cdot \sqrt{49})}$

Step 2.11

Subtract $9 i$ from $- 2 i$ .

$\frac{- 49}{(3 - 11 i) (i \cdot \sqrt{49})}$

Step 2.12

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

$\frac{- 49}{(3 - 11 i) (i \cdot \sqrt{7^{2}})}$

Step 2.13

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

$\frac{- 49}{(3 - 11 i) (i \cdot 7)}$

Step 2.14

Move $7$ to the left of $i$ .

$\frac{- 49}{(3 - 11 i) (7 \cdot i)}$

Step 2.15

Apply the distributive property.

$\frac{- 49}{3 (7 i) - 11 i (7 i)}$

Step 2.16

Multiply $7$ by $3$ .

$\frac{- 49}{21 i - 11 i (7 i)}$

Step 2.17

Multiply $- 11 i (7 i)$ .

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Step 2.17.1

Multiply $7$ by $- 11$ .

$\frac{- 49}{21 i - 77 i i}$

Step 2.17.2

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

$\frac{- 49}{21 i - 77 (i^{1} i)}$

Step 2.17.3

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

$\frac{- 49}{21 i - 77 (i^{1} i^{1})}$

Step 2.17.4

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

$\frac{- 49}{21 i - 77 i^{1 + 1}}$

Step 2.17.5

Add $1$ and $1$ .

$\frac{- 49}{21 i - 77 i^{2}}$

Step 2.18

Simplify each term.

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Step 2.18.1

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

$\frac{- 49}{21 i - 77 \cdot - 1}$

Step 2.18.2

Multiply $- 77$ by $- 1$ .

$\frac{- 49}{21 i + 77}$

Step 2.19

Reorder $21 i$ and $77$ .

$\frac{- 49}{77 + 21 i}$