Basic Math Examples

$\frac{1}{i^{43}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $i^{43}$ as ${(i^{4})}^{10} (i^{2} \cdot i)$ .

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Step 1.1.1

Factor out $i^{40}$ .

$\frac{1}{i^{40} i^{3}}$

Step 1.1.2

Rewrite $i^{40}$ as ${(i^{4})}^{10}$ .

$\frac{1}{{(i^{4})}^{10} i^{3}}$

Step 1.1.3

Factor out $i^{2}$ .

$\frac{1}{{(i^{4})}^{10} (i^{2} \cdot i)}$

Step 1.2

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

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Step 1.2.1

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

$\frac{1}{{({(i^{2})}^{2})}^{10} (i^{2} \cdot i)}$

Step 1.2.2

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

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

Step 1.2.3

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

$\frac{1}{1^{10} (i^{2} \cdot i)}$

Step 1.3

One to any power is one.

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

$\frac{1}{- i}$

Step 2

Cancel the common factor of $1$ and $- 1$ .

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Step 2.1

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

$\frac{- 1 (- 1)}{- i}$

Step 2.2

Move the negative in front of the fraction.

$- \frac{1}{i}$

Step 3

Multiply the numerator and denominator of $\frac{1}{i}$ by the conjugate of $i$ to make the denominator real.

$- (\frac{1}{i} \cdot \frac{i}{i})$

Step 4

Multiply.

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Step 4.1

Combine.

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

Step 4.2

Multiply $i$ by $1$ .

$- \frac{i}{i i}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

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

$- \frac{i}{i^{1} i}$

Step 4.3.2

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

$- \frac{i}{i^{1} i^{1}}$

Step 4.3.3

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

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

Step 4.3.4

Add $1$ and $1$ .

$- \frac{i}{i^{2}}$

Step 4.3.5

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

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

Step 5

Move the negative one from the denominator of $\frac{i}{- 1}$ .

$- (- 1 \cdot i)$

Step 6

Rewrite $- 1 \cdot i$ as $- i$ .

$- - i$

Step 7

Multiply $- 1$ by $- 1$ .

$1 i$

Step 8

Multiply $i$ by $1$ .

$i$