Express the listed radical in its minimal expression

Question

Problem 1: Use the prime decomposition of integers to express the listed radical in its minimal expression. Do not use decimals.

a. √(220500)

b. ³√(68600)

c. ³√(3267)√(4400)

d. √(1125)√(2420)

Problem 2: Fill in the table below. Note that you should refer to the section titled "Intervals" on pages 337-338 of the textbook and to Table 1 on page 338.

Interval      Inequality       Representation on the real line

                     x ≤ -1

                                                     °→

                      √3

(-3, Π )

                    3/5 ≤ x

Problem 3: Give the intervals that correspond to the inequalities listed below.

a. All real numbers x so that x ≥ - 2

b. All real numbers x so that 0 < x ≤ 8/3 and x ≤ 2

c. All real numbers x so that x > 3 or -6 ≤ x

Problem 4: In each of the following exercises, rewrite and simplify the given expression. Give your answer using positive exponents only.

a. (5x³yz²/xy²z)^-3/2

b. (uv²w³ - 3u²w)2(v^-2)

c. (√x³ + x²y^-2z)(xy²z³)

Problem 5: In each of the following exercises, expand and simplify.

a. √2(3x - √2x² + 1) - √18(1 - 4x)³

b. (t - u)² + 5(3t - u + 4u²)(1 + u)

c. (1 - 3x + x²)³(2 - 2x²)

Problem 6: In each of the following exercises, factor the given expressions.

a. 2y³ + 6y² + y + 3

b. 3x² - 18xy + 24y²

c. 50x³ + 20x² + 2x

Problem 7: In each of the following exercises, factor and simplify the terms, and then do the indicated operations.

a. 1/9x² - y² - (12x² - 10xy + 2y²)/(9x² - 6xy + y²)

b. √(x² + 5x + 4)/(x² + 8x + 16) - (x² - 3x - 4)/(x² - 16)

c. ((9x³ + 6x² + x)/27 x³ + 1) (6x - 1)/(3x² + x))

Problem 8: Determine which of the equations given below have real solutions and give the solutions of those that do.

a. 2x² + 3x = 6

b. x⁴ + 6x² - 3 = 0

Hint. Set a = x² and express the equation in terms of a.

c. 3x = 12x² - 5x

d. (x² + 7x)/3 = -5/2

e. (3x² + 3)/2 = x

Problem 9:

a. Rationalize the numerator of (√2 -6)/(5 + √2)

b. Rationalize the denominator of (5x - 2)/(√(2+x) -√6x)

c. Rationalize the denominator of (√8xy³ + 5√y)/(2y - √y)

Problem 10: Convert from radians to degrees the numbers given below. Note that you should refer to the section titled "Angles" on pages 358-359 of the textbook.

a. Π/5 + 3Π/8

b. 7Π/4 - √2Π

Problem 11: Convert from degrees to radians for the numbers given below.

a. (38/3)°

b. (90/5)°

Problem 12: Give the exact value of

a. tan (7Π/6)

b. cos (5Π/8) sin(5Π/8)

c. cos2 (Π/8)

Problem 13: In Unit 1 of the study guide we defined the trigonometric functions using a straight triangle with hypotenuse 1. Use similar triangles to define in any right-angle triangle with hypotenuse z the trigonometric functions as

cosθ = x/z sinθ = y/z tanθ = y/x

Hint. The circle below has radius 1.

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