To simplify an expression with exponents, first simplify each term
according to multiplication, division, distribution, and power to
power rules. Then, combine like terms and arrange the terms, putting
those with variables first, in order of highest exponent. Sometimes
like terms are only evident when one rewrites some of the terms using
the "power of a power" rule.
Example 1: Simplify (5x)^{2} + (x^{3})^{2} + (5x)(2x^{3}).
(5x)^{2} + (x^{3})^{2} + (5x)(2x^{3})
= 5^{2}x^{2} + x^{3(2)} + (5)(2)x^{1+3}
= 25x^{2} + x^{6} +10x^{4}
= x^{6} +10x^{4} +25x^{2}
Example 2: Simplify + (x^{3})^{7} -2x^{2} +2x^{3}.
+ (x^{3})^{7} -2x^{2} +2x^{3}
= x^{6-8} + x^{3(7)} -2x^{2} +2x^{3}
= 5x^{-2} + x^{2}1 - 2x^{2} +2x^{3}
= x^{2}1 + 2x^{3} -2x^{2} +5x^{-2}
Example 3: Simplify 2x^{2} +6x^{3} -3x^{3} + (4xy)^{2} + (5 + x)^{3} -
2x^{2} +6x^{3} -3x^{3} + (4xy)^{2} + (5 + x)^{3} -
= 2x^{2} +3x^{3} +4^{2}x^{2}y^{2} + (5 + x)^{3} - x^{3}
= 16x^{2}y^{2} + (5 + x)^{3} +2x^{3} +2x^{2}
Example 4: Simplify (x^{2} +2x^{2})^{5} - (3x^{3})(4x^{4}) + (11x)^{2}
(x^{2} +2x^{2})^{5} - (3x^{3})(4x^{4}) + (11x)^{2}
= (3x^{2})^{5} - (3)(4)x^{3+4} +11^{2}x^{2}
= 3^{5}x^{2(5)} -12x^{7} +121x^{2}
= 243x^{1}0 - 12x^{7} +121x^{2}
Example 5: Simplify (x + 3)^{5} + x^{5}x^{4}x^{3} -
(x + 3)^{5} + x^{5}x^{4}x^{3} -
= (x + 3)^{5} + x^{5+4+3} - (4x)^{3-1}
= (x + 3)^{5} + x^{1}2 - (4x)^{2}
= (x + 3)^{5} + x^{1}2 - 4^{2}x^{2}
= (x + 3)^{5} + x^{1}2 - 16x^{2}
= x^{1}2 + (x + 3)^{5} -16x^{2}