1. Simplify to an answer containing no negative exponents: [((4xy^-1)³)/((2x^-2y)² )].

(4xy-1)3 (2x-2y)2 = 43 x3 (y-1)3 22(x-2)2y2 = 43 x3 y-1 ·3 4x-2 ·2y2 = 42 x3 y-3 x-4y2 = 42 x3x4 y3 y2 = 42 x7 y5 .

2. Extract squares: Ö[(4(x + 1)²)]. Extract cubes: Ö[3]8(x + 1)³.

_______ Ö4(x + 1)2   = 2|x+1|, because nonnegativity is implicit in even radical symbol. Ö[3]8(x + 1)2 = 2(x+1), because odd roots are unique.

3. Solve: x² = x⁴.

x4 - x2 = 0 Þ x2(x+1)(x-1) = 0. x = 0 or x = 1 or x = -1.

4. Find the domain of: Ö[(x + 5 )] -[1/(x - 1)] .

No even rooting negative Þ x+ 5 ³ 0. No division by 0 Þ x \not = 1. -5 £ x < 1 or 1 < x.

5. A paints 1 room in 2 days. B paints 1 room in 5 days. If they work together, how many days does it take them to paint in 5 rooms?

(x days ) æ ç è 1 room 2 days + 1 room 5 days ö ÷ ø = 5 rooms . x = 5 1 2 + 1 5 .

6. Reduce: [(4 - x²)/(x² - 4x + 4)] .

4 - x2 x2 - 4x + 4 = (2-x)(2+x) (x-2)(x-2) = - x +2 x-2 .

7. Factor into irreducible factors: 7(x² - 2x + 1) - 3(x-1).

7(x2 - 2x + 1) - 3(x-1) = 7(x-1)(x-1) - 3(x-1) = (7(x-1) - 3)(x-1) = (7x - 10)(x-1).

8. Find the quotient and remainder (x⁴ + 2) ¸(3x³ - 1).

x4 3x3 = 1 3 x. (x4 + 2)- ( 1 3 x (3x3 - 1)) = 1 3 x + 2 = Remainder. Quotient = 1 3 x .

9. Write the product (x-2)²(x+2)² as a polynomial in standard form.

(x-2)2(x+2)2 = (x-2)(x+2)(x-2)(x+2) = (x2 - 4) (x2 - 4) = x4 - 8x2 + 16.

10. Solve for the exact answer: Ö3(x + 2) + 1 = 2 x - Ö2.

Isolate: Ö3 x - 2x = - 2 Ö3 - Ö2 -1. Factor: (Ö3 - 2)x = - 2 Ö3 - Ö2 -1 . x = - 2 Ö3 - Ö2 -1 Ö3 - 2 .