# Simplify Algebraic Expressions with Exponents

A set of intermediate algebra problems, related to simplifying algebraic expressions using the rules and properties of exponents. The solutions to these problems are at the bottom of the page.

Note: all letters (variables) used in the expressions below represent numbers that are not equal to 0.

1. Simplify the following expressions and write them with positive exponents.
a) x 2 x 3
b) a -2 a -3
c) a 5 / a 2
d) a -3 / a 7
e) (x 2) 3
f) (x -2) 5
g) (x 2 y 3) 3
h) (x -2 y 3) 8

2. Simplify the following expressions and write them with positive exponents.
a) ( y 4 / y 2 ) 3
b) ( y 4 / y -3 ) -2
c) ( y 4 x 12 / y 2 ) 3
d) ( y 12 x 56 / y 3 ) 0
e) (x 2 y 4) 2 / (x y -2 ) 3

3. Simplify the following expressions and write them with positive exponents.
a) ( x y 2 ) 2 ( x 2 y 2 ) 2 / ( x y ) 4
b) ( (a b 2) 3 (a 2 b 2) 4 ) 2
c) ( x 2 y 3 ) 0 ( x 4 y 3 ) 4 / ( x y 2 ) 3
d) ( x 3 y -2 ) 2 ( x -2 y 3 ) 4 / ( x -2 y -2 ) 3

1. a) x 2 x 3
= x 2 + 3 = x 5
b) a -2 a -3
= a -2 -3
= a -5 = 1 / a 5
c) a 5 / a 2
= a 5-2 = a 3
d) a -3 / a 7
= a -3 a -7
= a -10 = 1 / a 10
e) (x 2) 3
= x 2 * 3 = x 6
f) (x -2) 5
= x -2 * 5
= x -10 = 1 / x 10
g) (x 2 y 3) 3
= x 2 * 3 y 3 * 3 = x 2 * 3 y 3 * 3
h) (x -2 y 3) 8
= x -2*8 y 3*8 = x - 16 y 24 = y 24 / x 16
2. a) ( y 4 / y 2 ) 3
= y 4*3 / y 2*3 = y 12 / y 6
b) ( y 4 / y -3 ) -2
= y 4*(-2) / y -3*(-2) = y -8 / y 6 = 1 / y 14
c) ( y 4 x 12 / y 2 ) 3
= ( y 4 x 12) 3 / y 2*3 = y 4*3 x 12*3 / y 6 = y 12 x 36 / y 6 = y 6 x 36
d) ( y 12 x 56 / y 3 ) 0 = 1
e) (x 2 y 4) 2 / (x y -2 ) 3
= x 2*2 y 4*2 / (x 3 y -2*3 ) = x 4 y 8 / (x 3 y - 6 ) = x y 14
3. a) ( x y 2 ) 2 ( x 2 y 2 ) 2 / ( x y ) 4
= ( x 2 y 4 ) ( x 4 y 4 ) / ( x 4 y 4) = x 6 y 8 / ( x 4 y 4) = x 2 y 4
b) ( (a b 2) 3 (a 2 b 2) 4 ) 2
= (a b 2) 3*2 (a 2 b 2) 4*2 = (a b 2) 6 (a 2 b 2) 8 = a 6 b 2*6 a 2*8 b 2*8 = a 6 b 12 a 16 b 16 = a 22 b 28
c) ( x 2 y 3 ) 0 ( x 4 y 3 ) 4 / ( x y 2 ) 3
= 1 * x 4*4 y 3*4 / (x 3 y 2*3 ) = x 16 y 12 / (x 3 y 6) = x 13 y 6
d) ( x 3 y -2 ) 2 ( x -2 y 3 ) 4 / ( x -2 y -2 ) 3
= ( x 3*2 y -2*2 ) ( x -2*4 y 3*4 ) / ( x -2*3 y -2*3 ) = x -2 y 8 / ( x - 6 y - 6 ) = x 4 y 14

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