Unit III

Take Test: Unit III Assessment

Name	Unit III Assessment
Instructions	Instructions for math assessments: Mark the correct answer for each multiple-choice question in Blackboard. Please remember that textbooks and other reference materials can be used to complete this exam. On the free-response ("Essay") questions, please show and explain your work as best you can with the keyboard even if it is a one-step process. All free-response questions must show the complete process: step-by-step. Answers only on free-response questions result in zero points!!! So, before you start quizzes or exams, be sure that you have asked all the questions that you need to ask!! One more thing: Free-response questions on the exams are hand-graded by your instructor. You must show your work or give explanations to receive any credit. If you show your work and the process is reasonable, the instructor can give partial credit. Answers only result in zero points!!!
Multiple Attempts	Not allowed. This Test can only be taken once.
Force Completion	This Test can be saved and resumed later.

 Question 1

	Divide. 45 a -3 b10 by 5a -6 b 2 Answer
	9 a3 b8 40 a2 b5 9 a2 b5 40 a3 b8

4 points   explanation; 45/5 = 9

For the other parts, when you divide, subtract exponents.
a^-3 / a^-6 = a^3 (-3 (-) -6 = -3 +6 = 3)

 Question 2

	Divide 14a15 by 2a23 Answer
	7 a5 7/ a8 12 a5 12 a12

4 points  

 Question 3

	Simplify. Write your answer with only positive exponents. b -16/ b -4 Answer
	1/ b4 1/ b12 b12 b4

4 points   explanatiaon; b^-16 / b^-4 <---make both exponents positive. When an exponent is a negative number, move them from the numerator to the denominator, and vice versa...keeping the same exponent but eliminating the negative sign
b^4 / b^16 <--- subtract the exponents. 16-4 = 12
1 / b^12

 Question 4

	Evaluate 3√64 [cube root of 64] if possible Answer
	4 -8 - 64 / 3 Not a real number

4 points   explanation; cube root -64 = cube root (-4)^3 = -4
cube root 64 = cube root (4)^3 = 4

(the cube root and third power would cancel each other out.)

square root 64 = square root (8^2) = 8

again, the square root and second power cancel out.

square root -64 = no real solution, because there is no real number that multiplied by itself will give you a negative number. (negative times a negative gives you a positive.)

(If you know imaginary numbers, then the answer is 8i, but I don't think you do judging by your question.)

 

 Question 5

	A parsec is approximately 3.1 x 1016meters. If the distance to the farthest observable galaxies is approximately 1.62 x 109 parsecs, approximately how many meters is it to the farthest observable galaxies? Answer
	5 x 105 meters 2 x 1025 meters 5 x 1025 meters 2 x 105 meters

4 points   explanation; 1 parsec = 3.1 x 10^16 m
so 1.62 x 10^9 parsecs = 1.62 x 10^9 x 3.1 x 10^16
= 5.022 x 10^25 m
The farthest observable galaxies is 5.022 x 10^25 m
(approximately)

 Question 6

	Multiply. Write the answer in scientific notation. (2.4 x 10-5)(4 x 10-4) Answer
	9.6 x 1020 9.6 x 10-10 4.4 x 10-11 9.6 x 10-9

4 points   explanation; (2.4 x 10^–5)(4 x 10^–4)
= 2.4 x 4X10^–5 x 10^–4
= 9.6 X 10^-9

 Question 7

	Evaluate √-4 [square root of -4] if possible. Answer
	2 -2 2 or -2 Not a real number

4 points   explanation; Square roots of negative numbers
Negative numbers have square roots that lie outside the real numbers: Multiplying a real number by itself always results in a positive number. The square roots of negative numbers involve what are called imaginary numbers:

which for example leads to
The statements in "Two square roots for each number" also apply to negative numbers


square root of - 4 = square root of 4 times square root of -1

square root of 4 = 2
square root of -1 = i

therefore, square root of - 4 = 2i

 Question 8

	Evaluate 3 -5 Answer
	–15 –243 1/243 – 1/243

4 points  

 Question 9

	Simplify. Assume all variables represent positive numbers. √ (x4 y3) Answer
	x2y √y x2y√(xy) x4y2√y xy √(xy)

4 points  

 Question 10

	Multiply. a4b2 times ab3 Answer
	ab6 a5b5 ab5 a4b5

4 points  

 Question 11

	Simplify without negative exponents [ x y 5 / x -4 y 2 ] -2 Answer [ x y^5 / x^-4 y^2 ]^-2 = x^-2 y^-10 / x^8 y^-4 = y^4 / x^8 x^2 y^10 = y^4 / x^10 y^10 = 1/ x^10 y^6

10 points  

 Question 12

	Evaluate if possible 3√ - 64 [cube root of -64] and 3√ 64 [cube root of 64] and √ -64 [square root(-64)] and √ 64 [square root(64) ] Answer   cube root -64 = cube root (-4)^3 = -4 cube root 64 = cube root (4)^3 = 4 (the cube root and third power would cancel each other out.) square root 64 = square root (8^2) = 8 again, the square root and second power cancel out. square root -64 = no real solution, because there is no real number that multiplied by itself will give you a negative number. (negative times a negative gives you a positive.)

10 points  

 Question 13

	Simplify √ (32x5y2) that is sqrt (32x5y2) √ (32x5y2)=√16*2x^5y^2=4x^2y√(2x) Answer

10 points  

 Question 14

	Suppose a population of initial size 100 grows at the rate of 8% per year forever. What is the size of the population at the end of year 1? What is the size of the population at the end of year 2? What is the size of the population at the end of year 3? What is the size of the population at the end of year n (for any integer n)? What algebraic equation would you need to solve to find the number of years x that it would take for our population to reach 200? Use a calculator to solve to x. Answer a is the starting point (100) b is the growth factor (1.08) because the growth factor = growth rate +1, and in this case the growth rate is .08, or 8%, 1+.08= 1.08 x is the years ( for example, two years, 3, etc.) So.. to find the sizes of the populations plug in the year for x. Year 1: y=100(1.08)^1 = 108 Year 2: y=100(1.08)^2 = 116.64 Year 3: y=100(1.08)^3 = 125.97 So for n it would just be the basic equation, with n plugged in as x.. y=100(1.08)^n To find the years it will take the population to reach 200, you could keep plugging in years or use a graphing calculator and plug in the equation

10 points  

 Question 15

	The Schwarzschild radius describes the critical value to which the radius of a massive body must be reduced for it to become a black hole. R = 2 G M / c 2 where G = gravitational constant 6.7x10 -11 M= mass of the object C = speed of light 3x10 8 The sun has M = 2x10 30 . What is the Schwarzschild radius for the sun? [Note its true radius is 700,000.] Answer   R = 2 (6.7 x 10^-11) (2 x 10^30) / (3 x 10^8)² = (2 x 2 x 6.7) (10^-11x 10^30) / (9 x 10^16) = 26.8 x 10^19 / 9 x 10^16 = 2.978 x 10^3 = 2978 km

10 points  

 Question 16

	Simplify (– 2a2b3)2 / ( 2a2b)3 Answer   (– 2a^2b^3)^2 / ( 2a^2^b)^3: Apply the exponents to each term within the parentheses: (4a^4b^6) / (8a^6b^3) 4/8 = 1/2 a^4/a^6 = 1/(a^2) b^6/b^3 = b^3 You get (b^3) / (2a^2)

10 points  

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