Simplify i^40i^13i^37

$i^{40} \cdot i^{13} \cdot i^{37}$

Step 1

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

${(i^{4})}^{10} \cdot i^{13} \cdot i^{37}$

Step 2

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

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

${({(i^{2})}^{2})}^{10} \cdot i^{13} \cdot i^{37}$

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

${({(- 1)}^{2})}^{10} \cdot i^{13} \cdot i^{37}$

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

$1^{10} \cdot i^{13} \cdot i^{37}$

$1^{10} \cdot i^{13} \cdot i^{37}$

Step 3

One to any power is one.

$1 \cdot i^{13} \cdot i^{37}$

Step 4

Multiply $i^{13}$ by $1$ .

$i^{13} \cdot i^{37}$

Step 5

Rewrite $i^{13}$ as ${(i^{4})}^{3} i$ .

Factor out $i^{12}$ .

$i^{12} i \cdot i^{37}$

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

${(i^{4})}^{3} i \cdot i^{37}$

${(i^{4})}^{3} i \cdot i^{37}$

Step 6

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

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

${({(i^{2})}^{2})}^{3} i \cdot i^{37}$

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

${({(- 1)}^{2})}^{3} i \cdot i^{37}$

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

$1^{3} i \cdot i^{37}$

$1^{3} i \cdot i^{37}$

Step 7

One to any power is one.

$1 i \cdot i^{37}$

Step 8

Multiply $i$ by $1$ .

$i \cdot i^{37}$

Step 9

Rewrite $i^{37}$ as ${(i^{4})}^{9} i$ .

Factor out $i^{36}$ .

$i \cdot (i^{36} i)$

Rewrite $i^{36}$ as ${(i^{4})}^{9}$ .

$i \cdot ({(i^{4})}^{9} i)$

$i \cdot ({(i^{4})}^{9} i)$

Step 10

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

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

$i \cdot ({({(i^{2})}^{2})}^{9} i)$

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

$i \cdot ({({(- 1)}^{2})}^{9} i)$

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

$i \cdot (1^{9} i)$

$i \cdot (1^{9} i)$

Step 11

One to any power is one.

$i \cdot (1 i)$

Step 12

Multiply $i$ by $1$ .

$i \cdot i$

Step 13

Multiply $i i$ .

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

$i^{1} i$

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

$i^{1} i^{1}$

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

$i^{1 + 1}$

Add $1$ and $1$ .

$i^{2}$

$i^{2}$

Step 14

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

$- 1$

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