Finite Math Examples

$\frac{i^{4} + i^{9} + i^{16}}{2 + i^{5} + i^{10} + i^{15}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $i^{4}$ as $1$ .

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Step 1.1.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$\frac{{(i^{2})}^{2} + i^{9} + i^{16}}{2 + i^{5} + i^{10} + i^{15}}$

Step 1.1.2

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

$\frac{{(- 1)}^{2} + i^{9} + i^{16}}{2 + i^{5} + i^{10} + i^{15}}$

Step 1.1.3

Raise $- 1$ to the power of $2$ .

$\frac{1 + i^{9} + i^{16}}{2 + i^{5} + i^{10} + i^{15}}$

Step 1.2

Rewrite $i^{9}$ as ${(i^{4})}^{2} i$ .

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Step 1.2.1

Factor out $i^{8}$ .

$\frac{1 + i^{8} i + i^{16}}{2 + i^{5} + i^{10} + i^{15}}$

Step 1.2.2

Rewrite $i^{8}$ as ${(i^{4})}^{2}$ .

$\frac{1 + {(i^{4})}^{2} i + i^{16}}{2 + i^{5} + i^{10} + i^{15}}$

Step 1.3

Rewrite $i^{4}$ as $1$ .

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Step 1.3.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$\frac{1 + {({(i^{2})}^{2})}^{2} i + i^{16}}{2 + i^{5} + i^{10} + i^{15}}$

Step 1.3.2

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

$\frac{1 + {({(- 1)}^{2})}^{2} i + i^{16}}{2 + i^{5} + i^{10} + i^{15}}$

Step 1.3.3

Raise $- 1$ to the power of $2$ .

$\frac{1 + 1^{2} i + i^{16}}{2 + i^{5} + i^{10} + i^{15}}$

Step 1.4

One to any power is one.

$\frac{1 + 1 i + i^{16}}{2 + i^{5} + i^{10} + i^{15}}$

Step 1.5

Multiply $i$ by $1$ .

$\frac{1 + i + i^{16}}{2 + i^{5} + i^{10} + i^{15}}$

Step 1.6

Rewrite $i^{16}$ as ${(i^{4})}^{4}$ .

$\frac{1 + i + {(i^{4})}^{4}}{2 + i^{5} + i^{10} + i^{15}}$

Step 1.7

Rewrite $i^{4}$ as $1$ .

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Step 1.7.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$\frac{1 + i + {({(i^{2})}^{2})}^{4}}{2 + i^{5} + i^{10} + i^{15}}$

Step 1.7.2

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

$\frac{1 + i + {({(- 1)}^{2})}^{4}}{2 + i^{5} + i^{10} + i^{15}}$

Step 1.7.3

Raise $- 1$ to the power of $2$ .

$\frac{1 + i + 1^{4}}{2 + i^{5} + i^{10} + i^{15}}$

Step 1.8

One to any power is one.

$\frac{1 + i + 1}{2 + i^{5} + i^{10} + i^{15}}$

Step 1.9

Add $1$ and $1$ .

$\frac{2 + i}{2 + i^{5} + i^{10} + i^{15}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor out $i^{4}$ .

$\frac{2 + i}{2 + i^{4} i + i^{10} + i^{15}}$

Step 2.2

Rewrite $i^{4}$ as $1$ .

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Step 2.2.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$\frac{2 + i}{2 + {(i^{2})}^{2} i + i^{10} + i^{15}}$

Step 2.2.2

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

$\frac{2 + i}{2 + {(- 1)}^{2} i + i^{10} + i^{15}}$

Step 2.2.3

Raise $- 1$ to the power of $2$ .

$\frac{2 + i}{2 + 1 i + i^{10} + i^{15}}$

Step 2.3

Multiply $i$ by $1$ .

$\frac{2 + i}{2 + i + i^{10} + i^{15}}$

Step 2.4

Rewrite $i^{10}$ as ${(i^{4})}^{2} i^{2}$ .

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Step 2.4.1

Factor out $i^{8}$ .

$\frac{2 + i}{2 + i + i^{8} i^{2} + i^{15}}$

Step 2.4.2

Rewrite $i^{8}$ as ${(i^{4})}^{2}$ .

$\frac{2 + i}{2 + i + {(i^{4})}^{2} i^{2} + i^{15}}$

Step 2.5

Rewrite $i^{4}$ as $1$ .

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Step 2.5.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$\frac{2 + i}{2 + i + {({(i^{2})}^{2})}^{2} i^{2} + i^{15}}$

Step 2.5.2

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

$\frac{2 + i}{2 + i + {({(- 1)}^{2})}^{2} i^{2} + i^{15}}$

Step 2.5.3

Raise $- 1$ to the power of $2$ .

$\frac{2 + i}{2 + i + 1^{2} i^{2} + i^{15}}$

Step 2.6

One to any power is one.

$\frac{2 + i}{2 + i + 1 i^{2} + i^{15}}$

Step 2.7

Multiply $i^{2}$ by $1$ .

$\frac{2 + i}{2 + i + i^{2} + i^{15}}$

Step 2.8

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

$\frac{2 + i}{2 + i - 1 + i^{15}}$

Step 2.9

Rewrite $i^{15}$ as ${(i^{4})}^{3} (i^{2} \cdot i)$ .

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Step 2.9.1

Factor out $i^{12}$ .

$\frac{2 + i}{2 + i - 1 + i^{12} i^{3}}$

Step 2.9.2

Rewrite $i^{12}$ as ${(i^{4})}^{3}$ .

$\frac{2 + i}{2 + i - 1 + {(i^{4})}^{3} i^{3}}$

Step 2.9.3

Factor out $i^{2}$ .

$\frac{2 + i}{2 + i - 1 + {(i^{4})}^{3} (i^{2} \cdot i)}$

Step 2.10

Rewrite $i^{4}$ as $1$ .

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Step 2.10.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$\frac{2 + i}{2 + i - 1 + {({(i^{2})}^{2})}^{3} (i^{2} \cdot i)}$

Step 2.10.2

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

$\frac{2 + i}{2 + i - 1 + {({(- 1)}^{2})}^{3} (i^{2} \cdot i)}$

Step 2.10.3

Raise $- 1$ to the power of $2$ .

$\frac{2 + i}{2 + i - 1 + 1^{3} (i^{2} \cdot i)}$

Step 2.11

One to any power is one.

$\frac{2 + i}{2 + i - 1 + 1 (i^{2} \cdot i)}$

Step 2.12

Multiply $i^{2} \cdot i$ by $1$ .

$\frac{2 + i}{2 + i - 1 + i^{2} \cdot i}$

Step 2.13

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

$\frac{2 + i}{2 + i - 1 - 1 \cdot i}$

Step 2.14

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

$\frac{2 + i}{2 + i - 1 - i}$

Step 2.15

Subtract $1$ from $2$ .

$\frac{2 + i}{1 + i - i}$

Step 2.16

Subtract $i$ from $i$ .

$\frac{2 + i}{1 + 0}$

Step 2.17

Add $1$ and $0$ .

$\frac{2 + i}{1}$

Step 3

Divide $2 + i$ by $1$ .

$2 + i$