# Exponents

### Simplifying Expressions with Exponents

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)
= 52 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 21 - 2x 2 +2x 3
= x 21 + 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 +42 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 +112 x 2
= 35 x 2(5) -12x 7 +121x 2
= 243x 10 - 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 12 - (4x)2
= (x + 3)5 + x 12 - 42 x 2
= (x + 3)5 + x 12 - 16x 2
= x 12 + (x + 3)5 -16x 2

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