Integer Exponent Problems worksheet from Fall 2004, featuring 54 exercises to simplify expressions with integer exponents.

The user wants me to solve all 54 integer exponent problems from the worksheet. Let me work through each one carefully.

Let me go through each problem:

(1) 5² = 25
(2) 3⁵ = 243
(3) x²x³ = x
(4) x⁻² = 1/x²
(5) 2⁻³ = 1/8
(6) (x²)³ = x⁶
(7) 3²2⁵ = 9·32 = 288
(8) 1/x⁻³ = x³


(9) 6⁻¹ = 1/6
(10) 2x⁴(x³) = 2x⁷
(11) (2y²)² = 4y
(12) x⁵/x³ = x²
(13) x/x = 1/x
(14) (x²y³)⁵ = x¹⁰y¹⁵
(15) -3² = -9
(16) 4/x⁻² = 4x²
(17) 3⁻² = 1/9
(18) (-3)² = 9
(19) (-2)³ = -8
(20) 5⁰ = 1
(21) r⁻²/-r² = -r⁻⁴ = -1/r⁴
(22) s⁻¹ = 1/s
(23) 5⁻² = 1/25
(24) 1/3⁻² = 9
(25) s⁻² = 1/s²
(26) z⁵z⁻⁴ = z
(27) (r³)⁻² = r⁶
(28) (x³)⁻² = x⁻⁶ = 1/x
(29) z²z⁴/z³ = z³
(30) (2x⁻²)³ = 8x⁻⁶ = 8/x⁶
(31) (2x⁻²)⁻³ = x⁶/8
(32) (3²)(2³) = 72
(33) 5²5⁻ = 5⁻³ = 1/125
(34) (2x²)(3xy²) = 6x³y²
(35) z⁻⁴/z⁻² = z⁻² = 1/z²
(36) x/2⁻³ = 8x
(37) 2y⁻³/y⁻⁴ = 2y
(38) 3x⁻³ = 3/x³
(39) (2/3)⁻¹ = 3/2
(40) (2/5)³ = 8/125
(41) (4/5)⁻² = 25/16
(42) (2x⁻³y²)⁻² = y⁻⁴/(4x) = 1/(4x⁶y)
(43) 7⁻² = 1/49
(44) -5⁻² = -1/25
(45) (-3)⁻² = 1/9
(46) (2z⁻¹)⁻¹ = z/2
(47) (r⁻²)⁻¹ = r²
(48) x⁻³/x⁻⁵ = x²
(49) (x²y⁻¹)⁻³ = y³/x⁶
(50) y⁻³x²/y²x⁻³ = x⁵/y
(51) 5⁻¹3² = 9/5
(52) z⁻³/3⁻² = 9/z³
(53) (x²y⁻²)⁴ = y⁸/x⁸
(54) 2⁻¹ + 3² = 1/2 + 9 = 19/2

Now I'm verifying a few of these calculations to make sure they're correct, particularly checking problem 21 where I need to be careful with the negative signs and exponents. I'm finishing up the verification of the remaining exponent problems, working through each one systematically to confirm the simplifications are correct, then compiling the final answers for all 54 problems.