Evaluate i^19i^23i^11*i^21

$i^{19} \cdot i^{23} \cdot i^{11} \cdot i^{21}$

Rewrite $i^{19}$ as ${(i^{4})}^{4} (i^{2} \cdot i)$ .

Factor out $i^{16}$ .

$i^{16} i^{3} \cdot i^{23} \cdot i^{11} \cdot i^{21}$

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

${(i^{4})}^{4} i^{3} \cdot i^{23} \cdot i^{11} \cdot i^{21}$

Factor out $i^{2}$ .

${(i^{4})}^{4} (i^{2} \cdot i) \cdot i^{23} \cdot i^{11} \cdot i^{21}$

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

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

${({(i^{2})}^{2})}^{4} (i^{2} \cdot i) \cdot i^{23} \cdot i^{11} \cdot i^{21}$

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

${({(- 1)}^{2})}^{4} (i^{2} \cdot i) \cdot i^{23} \cdot i^{11} \cdot i^{21}$

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

$1^{4} (i^{2} \cdot i) \cdot i^{23} \cdot i^{11} \cdot i^{21}$

One to any power is one.

$1 (i^{2} \cdot i) \cdot i^{23} \cdot i^{11} \cdot i^{21}$

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

$i^{2} \cdot i \cdot i^{23} \cdot i^{11} \cdot i^{21}$

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

$- 1 \cdot i \cdot i^{23} \cdot i^{11} \cdot i^{21}$

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

$- i \cdot i^{23} \cdot i^{11} \cdot i^{21}$

Rewrite $i^{23}$ as ${(i^{4})}^{5} (i^{2} \cdot i)$ .

Factor out $i^{20}$ .

$- i \cdot (i^{20} i^{3}) \cdot i^{11} \cdot i^{21}$

Rewrite $i^{20}$ as ${(i^{4})}^{5}$ .

$- i \cdot ({(i^{4})}^{5} i^{3}) \cdot i^{11} \cdot i^{21}$

Factor out $i^{2}$ .

$- i \cdot ({(i^{4})}^{5} (i^{2} \cdot i)) \cdot i^{11} \cdot i^{21}$

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

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

$- i \cdot ({({(i^{2})}^{2})}^{5} (i^{2} \cdot i)) \cdot i^{11} \cdot i^{21}$

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

$- i \cdot ({({(- 1)}^{2})}^{5} (i^{2} \cdot i)) \cdot i^{11} \cdot i^{21}$

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

$- i \cdot (1^{5} (i^{2} \cdot i)) \cdot i^{11} \cdot i^{21}$

One to any power is one.

$- i \cdot (1 (i^{2} \cdot i)) \cdot i^{11} \cdot i^{21}$

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

$- i \cdot (i^{2} \cdot i) \cdot i^{11} \cdot i^{21}$

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

$- i \cdot (- 1 \cdot i) \cdot i^{11} \cdot i^{21}$

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

$- i \cdot (- i) \cdot i^{11} \cdot i^{21}$

Multiply $- i (- i)$ .

Multiply $- 1$ by $- 1$ .

$1 i i \cdot i^{11} \cdot i^{21}$

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

$1 (i^{1} i) \cdot i^{11} \cdot i^{21}$

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

$1 (i^{1} i^{1}) \cdot i^{11} \cdot i^{21}$

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

$1 i^{1 + 1} \cdot i^{11} \cdot i^{21}$

Add $1$ and $1$ .

$1 i^{2} \cdot i^{11} \cdot i^{21}$

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

$i^{2} \cdot i^{11} \cdot i^{21}$

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

$- 1 \cdot i^{11} \cdot i^{21}$

Rewrite $i^{11}$ as ${(i^{4})}^{2} (i^{2} \cdot i)$ .

Factor out $i^{8}$ .

$- 1 \cdot (i^{8} i^{3}) \cdot i^{21}$

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

$- 1 \cdot ({(i^{4})}^{2} i^{3}) \cdot i^{21}$

Factor out $i^{2}$ .

$- 1 \cdot ({(i^{4})}^{2} (i^{2} \cdot i)) \cdot i^{21}$

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

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

$- 1 \cdot ({({(i^{2})}^{2})}^{2} (i^{2} \cdot i)) \cdot i^{21}$

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

$- 1 \cdot ({({(- 1)}^{2})}^{2} (i^{2} \cdot i)) \cdot i^{21}$

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

$- 1 \cdot (1^{2} (i^{2} \cdot i)) \cdot i^{21}$

One to any power is one.

$- 1 \cdot (1 (i^{2} \cdot i)) \cdot i^{21}$

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

$- 1 \cdot (i^{2} \cdot i) \cdot i^{21}$

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

$- 1 \cdot (- 1 \cdot i) \cdot i^{21}$

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

$- 1 \cdot (- i) \cdot i^{21}$

Multiply $- 1$ by $- 1$ .

$1 i \cdot i^{21}$

Multiply $i$ by $1$ .

$i \cdot i^{21}$

Rewrite $i^{21}$ as ${(i^{4})}^{5} i$ .

Factor out $i^{20}$ .

$i \cdot (i^{20} i)$

Rewrite $i^{20}$ as ${(i^{4})}^{5}$ .

$i \cdot ({(i^{4})}^{5} i)$

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

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

$i \cdot ({({(i^{2})}^{2})}^{5} i)$

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

$i \cdot ({({(- 1)}^{2})}^{5} i)$

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

$i \cdot (1^{5} i)$

One to any power is one.

$i \cdot (1 i)$

Multiply $i$ by $1$ .

$i \cdot i$

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 $1$ .

$i^{2}$

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

$- 1$

Evaluate i^19*i^23*i^11*i^21

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Evaluate i^19i^23i^11*i^21