Algebra Examples

$s = r (\frac{{(1 + i)}^{n} - 1}{i})$

Step 1

Rewrite the equation as $r (\frac{{(1 + i)}^{n} - 1}{i}) = s$ .

$r (\frac{{(1 + i)}^{n} - 1}{i}) = s$

Step 2

Simplify $r (\frac{{(1 + i)}^{n} - 1}{i})$ .

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Step 2.1

Multiply the numerator and denominator of $\frac{{(1 + i)}^{n} - 1}{i}$ by the conjugate of $i$ to make the denominator real.

$r (\frac{{(1 + i)}^{n} - 1}{i} \cdot \frac{i}{i}) = s$

Step 2.2

Multiply.

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Step 2.2.1

Combine.

$r \frac{({(1 + i)}^{n} - 1) i}{i i} = s$

Step 2.2.2

Simplify the numerator.

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Step 2.2.2.1

Apply the distributive property.

$r \frac{{(1 + i)}^{n} i - 1 i}{i i} = s$

Step 2.2.2.2

Rewrite $- 1 i$ as $- i$ .

$r \frac{{(1 + i)}^{n} i - i}{i i} = s$

Step 2.2.2.3

Reorder factors in ${(1 + i)}^{n} i - i$ .

$r \frac{i {(1 + i)}^{n} - i}{i i} = s$

Step 2.2.3

Simplify the denominator.

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Step 2.2.3.1

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

$r \frac{i {(1 + i)}^{n} - i}{i^{1} i} = s$

Step 2.2.3.2

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

$r \frac{i {(1 + i)}^{n} - i}{i^{1} i^{1}} = s$

Step 2.2.3.3

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

$r \frac{i {(1 + i)}^{n} - i}{i^{1 + 1}} = s$

Step 2.2.3.4

Add $1$ and $1$ .

$r \frac{i {(1 + i)}^{n} - i}{i^{2}} = s$

Step 2.2.3.5

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

$r \frac{i {(1 + i)}^{n} - i}{- 1} = s$

Step 2.3

Simplify the expression.

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Step 2.3.1

Move the negative one from the denominator of $\frac{i {(1 + i)}^{n} - i}{- 1}$ .

$r (- 1 \cdot (i {(1 + i)}^{n} - i)) = s$

Step 2.3.2

Rewrite $- 1 \cdot (i {(1 + i)}^{n} - i)$ as $- (i {(1 + i)}^{n} - i)$ .

$r (- (i {(1 + i)}^{n} - i)) = s$

Step 2.3.3

Rewrite using the commutative property of multiplication.

$- r (i {(1 + i)}^{n} - i) = s$

Step 2.4

Apply the distributive property.

$- r (i {(1 + i)}^{n}) - r (- i) = s$

Step 2.5

Rewrite using the commutative property of multiplication.

$- 1 \cdot i r {(1 + i)}^{n} - r (- i) = s$

Step 2.6

Multiply $- r (- i)$ .

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Step 2.6.1

Multiply $- 1$ by $- 1$ .

$- 1 \cdot i r {(1 + i)}^{n} + 1 r i = s$

Step 2.6.2

Multiply $r$ by $1$ .

$- 1 \cdot i r {(1 + i)}^{n} + r i = s$

Step 2.7

Rewrite $- 1 i$ as $- i$ .

$- i r {(1 + i)}^{n} + r i = s$

Step 3

Factor $i r$ out of $- i r {(1 + i)}^{n} + r i$ .

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Step 3.1

Factor $i r$ out of $- i r {(1 + i)}^{n}$ .

$i r (- 1 {(1 + i)}^{n}) + r i = s$

Step 3.2

Factor $i r$ out of $r i$ .

$i r (- 1 {(1 + i)}^{n}) + i r (1) = s$

Step 3.3

Factor $i r$ out of $i r (- 1 {(1 + i)}^{n}) + i r (1)$ .

$i r (- 1 {(1 + i)}^{n} + 1) = s$

Step 4

Rewrite $- 1 {(1 + i)}^{n}$ as $- {(1 + i)}^{n}$ .

$i r (- {(1 + i)}^{n} + 1) = s$

Step 5

Divide each term in $i r (- {(1 + i)}^{n} + 1) = s$ by $i (- {(1 + i)}^{n} + 1)$ and simplify.

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Step 5.1

Divide each term in $i r (- {(1 + i)}^{n} + 1) = s$ by $i (- {(1 + i)}^{n} + 1)$ .

$\frac{i r (- {(1 + i)}^{n} + 1)}{i (- {(1 + i)}^{n} + 1)} = \frac{s}{i (- {(1 + i)}^{n} + 1)}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $i$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{i r (- {(1 + i)}^{n} + 1)}{i (- {(1 + i)}^{n} + 1)} = \frac{s}{i (- {(1 + i)}^{n} + 1)}$

Step 5.2.1.2

Rewrite the expression.

$\frac{r (- {(1 + i)}^{n} + 1)}{- {(1 + i)}^{n} + 1} = \frac{s}{i (- {(1 + i)}^{n} + 1)}$

Step 5.2.2

Cancel the common factor of $- {(1 + i)}^{n} + 1$ .

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Step 5.2.2.1

Cancel the common factor.

$\frac{r (- {(1 + i)}^{n} + 1)}{- {(1 + i)}^{n} + 1} = \frac{s}{i (- {(1 + i)}^{n} + 1)}$

Step 5.2.2.2

Divide $r$ by $1$ .

$r = \frac{s}{i (- {(1 + i)}^{n} + 1)}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Apply the distributive property.

$r = \frac{s}{i (- {(1 + i)}^{n}) + i \cdot 1}$

Step 5.3.2

Simplify the expression.

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Step 5.3.2.1

Multiply $i$ by $1$ .

$r = \frac{s}{i (- {(1 + i)}^{n}) + i}$

Step 5.3.2.2

Reorder factors in $i (- {(1 + i)}^{n}) + i$ .

$r = \frac{s}{- i {(1 + i)}^{n} + i}$

Step 5.3.3

Factor $- 1$ out of $- i {(1 + i)}^{n}$ .

$r = \frac{s}{- (i {(1 + i)}^{n}) + i}$

Step 5.3.4

Factor $- 1$ out of $i$ .

$r = \frac{s}{- (i {(1 + i)}^{n}) - 1 (- i)}$

Step 5.3.5

Factor $- 1$ out of $- (i {(1 + i)}^{n}) - 1 (- i)$ .

$r = \frac{s}{- (i {(1 + i)}^{n} - i)}$

Step 5.3.6

Rewrite negatives.

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Step 5.3.6.1

Rewrite $- (i {(1 + i)}^{n} - i)$ as $- 1 (i {(1 + i)}^{n} - i)$ .

$r = \frac{s}{- 1 (i {(1 + i)}^{n} - i)}$

Step 5.3.6.2

Move the negative in front of the fraction.

$r = - \frac{s}{i {(1 + i)}^{n} - i}$