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.
![336_Circle-Trigonometric functions.jpg](https://secure.tutorsglobe.com/CMSImages/336_Circle-Trigonometric functions.jpg)
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