Simplify i^13i^29i^6

$i^{13} \cdot i^{29} \cdot i^{6}$

Step 1

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

Factor out $i^{12}$ .

$i^{12} i \cdot i^{29} \cdot i^{6}$

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

${(i^{4})}^{3} i \cdot i^{29} \cdot i^{6}$

${(i^{4})}^{3} i \cdot i^{29} \cdot i^{6}$

Step 2

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

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

${({(i^{2})}^{2})}^{3} i \cdot i^{29} \cdot i^{6}$

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

${({(- 1)}^{2})}^{3} i \cdot i^{29} \cdot i^{6}$

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

$1^{3} i \cdot i^{29} \cdot i^{6}$

$1^{3} i \cdot i^{29} \cdot i^{6}$

Step 3

One to any power is one.

$1 i \cdot i^{29} \cdot i^{6}$

Step 4

Multiply $i$ by $1$ .

$i \cdot i^{29} \cdot i^{6}$

Step 5

Rewrite $i^{29}$ as ${(i^{4})}^{7} i$ .

Factor out $i^{28}$ .

$i \cdot (i^{28} i) \cdot i^{6}$

Rewrite $i^{28}$ as ${(i^{4})}^{7}$ .

$i \cdot ({(i^{4})}^{7} i) \cdot i^{6}$

$i \cdot ({(i^{4})}^{7} i) \cdot i^{6}$

Step 6

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

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

$i \cdot ({({(i^{2})}^{2})}^{7} i) \cdot i^{6}$

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

$i \cdot ({({(- 1)}^{2})}^{7} i) \cdot i^{6}$

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

$i \cdot (1^{7} i) \cdot i^{6}$

$i \cdot (1^{7} i) \cdot i^{6}$

Step 7

One to any power is one.

$i \cdot (1 i) \cdot i^{6}$

Step 8

Multiply $i$ by $1$ .

$i \cdot i \cdot i^{6}$

Step 9

Multiply $i i$ .

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

$i^{1} i \cdot i^{6}$

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

$i^{1} i^{1} \cdot i^{6}$

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

$i^{1 + 1} \cdot i^{6}$

Add $1$ and $1$ .

$i^{2} \cdot i^{6}$

$i^{2} \cdot i^{6}$

Step 10

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

$- 1 \cdot i^{6}$

Step 11

Factor out $i^{4}$ .

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

Step 12

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

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

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

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

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

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

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

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

Step 13

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

$- 1 \cdot i^{2}$

Step 14

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

$- 1 \cdot - 1$

Step 15

Multiply $- 1$ by $- 1$ .

$1$

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