Simplify i^15i^27i^15*i^8

$i^{15} \cdot i^{27} \cdot i^{15} \cdot i^{8}$

Step 1

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

Factor out $i^{12}$ .

$i^{12} i^{3} \cdot i^{27} \cdot i^{15} \cdot i^{8}$

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

${(i^{4})}^{3} i^{3} \cdot i^{27} \cdot i^{15} \cdot i^{8}$

Factor out $i^{2}$ .

${(i^{4})}^{3} (i^{2} \cdot i) \cdot i^{27} \cdot i^{15} \cdot i^{8}$

Step 2

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

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

${({(i^{2})}^{2})}^{3} (i^{2} \cdot i) \cdot i^{27} \cdot i^{15} \cdot i^{8}$

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

${({(- 1)}^{2})}^{3} (i^{2} \cdot i) \cdot i^{27} \cdot i^{15} \cdot i^{8}$

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

$1^{3} (i^{2} \cdot i) \cdot i^{27} \cdot i^{15} \cdot i^{8}$

Step 3

One to any power is one.

$1 (i^{2} \cdot i) \cdot i^{27} \cdot i^{15} \cdot i^{8}$

Step 4

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

$i^{2} \cdot i \cdot i^{27} \cdot i^{15} \cdot i^{8}$

Step 5

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

$- 1 \cdot i \cdot i^{27} \cdot i^{15} \cdot i^{8}$

Step 6

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

$- i \cdot i^{27} \cdot i^{15} \cdot i^{8}$

Step 7

Rewrite $i^{27}$ as ${(i^{4})}^{6} (i^{2} \cdot i)$ .

Factor out $i^{24}$ .

$- i \cdot (i^{24} i^{3}) \cdot i^{15} \cdot i^{8}$

Rewrite $i^{24}$ as ${(i^{4})}^{6}$ .

$- i \cdot ({(i^{4})}^{6} i^{3}) \cdot i^{15} \cdot i^{8}$

Factor out $i^{2}$ .

$- i \cdot ({(i^{4})}^{6} (i^{2} \cdot i)) \cdot i^{15} \cdot i^{8}$

Step 8

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

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

$- i \cdot ({({(i^{2})}^{2})}^{6} (i^{2} \cdot i)) \cdot i^{15} \cdot i^{8}$

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

$- i \cdot ({({(- 1)}^{2})}^{6} (i^{2} \cdot i)) \cdot i^{15} \cdot i^{8}$

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

$- i \cdot (1^{6} (i^{2} \cdot i)) \cdot i^{15} \cdot i^{8}$

Step 9

One to any power is one.

$- i \cdot (1 (i^{2} \cdot i)) \cdot i^{15} \cdot i^{8}$

Step 10

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

$- i \cdot (i^{2} \cdot i) \cdot i^{15} \cdot i^{8}$

Step 11

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

$- i \cdot (- 1 \cdot i) \cdot i^{15} \cdot i^{8}$

Step 12

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

$- i \cdot (- i) \cdot i^{15} \cdot i^{8}$

Step 13

Multiply $- i (- i)$ .

Multiply $- 1$ by $- 1$ .

$1 i i \cdot i^{15} \cdot i^{8}$

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

$1 (i^{1} i) \cdot i^{15} \cdot i^{8}$

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

$1 (i^{1} i^{1}) \cdot i^{15} \cdot i^{8}$

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

$1 i^{1 + 1} \cdot i^{15} \cdot i^{8}$

Add $1$ and $1$ .

$1 i^{2} \cdot i^{15} \cdot i^{8}$

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

$i^{2} \cdot i^{15} \cdot i^{8}$

Step 14

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

$- 1 \cdot i^{15} \cdot i^{8}$

Step 15

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

Factor out $i^{12}$ .

$- 1 \cdot (i^{12} i^{3}) \cdot i^{8}$

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

$- 1 \cdot ({(i^{4})}^{3} i^{3}) \cdot i^{8}$

Factor out $i^{2}$ .

$- 1 \cdot ({(i^{4})}^{3} (i^{2} \cdot i)) \cdot i^{8}$

Step 16

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

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

$- 1 \cdot ({({(i^{2})}^{2})}^{3} (i^{2} \cdot i)) \cdot i^{8}$

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

$- 1 \cdot ({({(- 1)}^{2})}^{3} (i^{2} \cdot i)) \cdot i^{8}$

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

$- 1 \cdot (1^{3} (i^{2} \cdot i)) \cdot i^{8}$

Step 17

One to any power is one.

$- 1 \cdot (1 (i^{2} \cdot i)) \cdot i^{8}$

Step 18

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

$- 1 \cdot (i^{2} \cdot i) \cdot i^{8}$

Step 19

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

$- 1 \cdot (- 1 \cdot i) \cdot i^{8}$

Step 20

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

$- 1 \cdot (- i) \cdot i^{8}$

Step 21

Multiply $- 1$ by $- 1$ .

$1 i \cdot i^{8}$

Step 22

Multiply $i$ by $1$ .

$i \cdot i^{8}$

Step 23

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

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

Step 24

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

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

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

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

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

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

$i \cdot 1^{2}$

Step 25

One to any power is one.

$i \cdot 1$

Step 26

Multiply $i$ by $1$ .

$i$

Simplify i^15*i^27*i^15*i^8

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Simplify i^15i^27i^15*i^8