Algebra Examples

${(x y - 2 y^{- 2})}^{21}$

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{21} \frac{21!}{(21 - k)! k!} \cdot {(x y)}^{21 - k} \cdot {(- 2 y^{- 2})}^{k}$

Step 2

Expand the summation.

$\frac{21!}{(21 - 0)! 0!} \cdot {(x y)}^{21 - 0} \cdot {(- 2 y^{- 2})}^{0} + \frac{21!}{(21 - 1)! 1!} \cdot {(x y)}^{21 - 1} \cdot {(- 2 y^{- 2})}^{1} + \dots + \frac{21!}{(21 - 21)! 21!} \cdot {(x y)}^{21 - 21} \cdot {(- 2 y^{- 2})}^{21}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(x y)}^{21} \cdot {(- 2 y^{- 2})}^{0} + 21 \cdot {(x y)}^{20} \cdot {(- 2 y^{- 2})}^{1} + \dots + 1 \cdot {(x y)}^{0} \cdot {(- 2 y^{- 2})}^{21}$

Step 4

Simplify the polynomial result.

$x^{21} y^{21} - 42 x^{20} y^{18} + 840 x^{19} y^{15} - 10640 x^{18} y^{12} + 95760 x^{17} y^{9} - 651168 x^{16} y^{6} + 3472896 x^{15} y^{3} - 14883840 x^{14} + \frac{52093440 x^{13}}{y^{3}} - \frac{150492160 x^{12}}{y^{6}} + \frac{361181184 x^{11}}{y^{9}} - \frac{722362368 x^{10}}{y^{12}} + \frac{1203937280 x^{9}}{y^{15}} - \frac{1666990080 x^{8}}{y^{18}} + \frac{1905131520 x^{7}}{y^{21}} - \frac{1778122752 x^{6}}{y^{24}} + \frac{1333592064 x^{5}}{y^{27}} - \frac{784465920 x^{4}}{y^{30}} + \frac{348651520 x^{3}}{y^{33}} - \frac{110100480 x^{2}}{y^{36}} + \frac{22020096 x}{y^{39}} - \frac{2097152}{y^{42}}$