Algebra Questions

EXPONENTS a a 5 so Estimating Square Roots A number is called a if its square root is an integer For example 25 and 36 are both perfect squares since 25 52 and 36 6 so 25 5 and 36 6 If we don t know the square root of some number x and don t have a calculator handy we can estimate the square root by bracketing it betweer two perfect squares for which we do know the square roots For example to estimate IO we know and 10 10 8 is between 2 Estimate the following square 23 Therefore 10 lies somewhere between and closer to 9 than to 16 According to a calculator 10 and 642 roots where and are perfect squares probably closer to because 10 is 95 is between and

EX 3 Find the value of each expression A log 32 B log 18

EX 2 Expand each logarithm 8x 5y A log c log 9x log 37 B loga D log 4

L Simplify the following expressions and express your answers with positive exponents a x x 5 d 13 8 JAL M Before we can apply the rules of exponents to expressions in the form of a n where the exponent is a fraction we need to consider what such an expression means Simplify the following expressions a x1 2 x 1 2 c m 2 m 2 9 then m d h 2 h 2 If h The expression a called a b a xxx h1 2 Simplify the following expressions 1 2 is often written as If h 5 5 6 d h 3 h 3 h1 3 1 2 c 5 5 5 5 then x h1 3 1 3 C 1 2 16 broc com what s another way to write h 23 x 2 and is called the ola 1 3 The nth Root If a is a real number and n is a positive integer a a b 41 2 41 2 what s another way to write h 32 b 1 3 27 the nth root of a e v 2 w5 v of a The symbol 8 is

EX 1 Write each expression as a single logarithre A log 32 log 4 C 2 log 6 log 9 E

Determine whether this is a parabola x 2y 12y 9 O Parabola O Not a parabola If it is a parabola write the equation in standard form y k 4p x h or x h 4p y k Otherwise leave the rest blank h k P p

Answer Attempt 1 out of 2 Watch Video Question Members of a softball team raised 1525 50 to go to a tournament They rented a bus for 1075 50 and budgeted 37 50 per player for meals Write and solve an equation which can be used to determine x the number of players the team can bring to the tournament Equation Show Example

O Solve each equation 9 9 2n 2 10 8 A 3 0 10 C B 9 D 0 2 B Evaluate each using the values given 10 g m q q use m 6 and q 2 9

Prove by Insuction 1 2 3 4 2 2 4 1 n z 24 1

REVIEW Simplify each difference 1 1 4 2n n 3 6n 2n A 3n 4n 1 B 3n 19n 1 C 3m 16n 1 D 3n 8m 1 Simplify each sum 2 4x 6x 4r 3r A 10x 2x C 5x B 10x D 3x

E 16 17 18 E Evaluate each function ANSWER SHOULD BE A FRACTION OR MIXED NUMB ANSWERS GIVEN AS A DECINMAL WILL BE COUNTED AS INCORRECT 5 Mx x Find Solve each inequality 6 57m 1 6211 A 7 2 B 7 orm 2 C 2005 D 20 or 5 4 O Solve each inequality and graph its solution 7 3 2 4 4 45 A ke 1 ork 7 B 7 4 7 C 5 or 7 D 7 5 DO zoom in

6 Find the slope of each line GIVEN a graph or an equation 5 X Find the slope of each line 5 3y 9x 3 Find the slope of the line through each pair of points 7 18 3 5 7

REVIEW 2 Final Sum REVIEW Write the point slope form of the equation of the line through the given points 12 through 0 3 and 2 1 13 A y 3 x 3 B y 4 x 3 D j x 3 Evaluate each function 2 h x x 4x Find M 10 B S D 140 A 32 C 4 Evaluate each function Round to the nearest hundredth if necessary TWO DIGITS

Solve the following system of equations There is a unique solution which should be entered as the ordered triplet x y z 2x 5y z 25 4x4z 24 6x 15y 3x 27

Solve the following system of equations If there is a unique solution enter it as an ordered pair in the form x y If there is no solution enter Inconsistent If there are infinitely many solutions enter your answer as a line in the form y mx b 2x 3y 3 6x9y 9

By hand solve the system using matrices and row operations List each row operation as we have done in class i e R3 10r2 73 Give exact answers in fraction form If the system has no solution state this If the system has an infinite number of solutions give one solution x y z 8 2x 3y z 2 3x 2y9z 9

8 Two ocean liners leave from the same port in Puerto Rico at 10 00 a m One travels at a bearing of N 51 W at 11 miles per hour and the other travels at a bearing of S 53 W at 17 miles per hour Approximate the distance between them at noon the same day Round your answer to two decimal places O 20 28 miles O 39 93 miles O 35 75 miles O 30 25 miles

in the figure below a 10 b 11 and c 14 Use this information to solve the parallelogram for 0 The diagonals of the parallelogram are represented by c and d Round your answer to two decimal places C b a 8 P

0 Give the three formulas for the law of cosines and explain how you use the Law of Cosines Also tell which congruence cases use Law of Cosines and which use Law of Sines ex SAS SSS

4 Determine whether the Law of Sines or the Law of Cosines is needed to solve the triangle Then solve the triangle Round your answer to tw decimal places a 152 B 17 C 12 Law of Sines A 151 b 65 19 c 91 67 Law of Sines No solution Law of Sines A 151 b 91 67 c 65 19 Law of Cosines A 151 b 91 67 c 65 19

2 Use the low of Cosines to solve the given triangle Round your answer to two decimal places a 54 b 24 c 71 0 A 126 69 B 15 73 C 37 58 0 A 37 58 B 126 69 C 15 73 0 A 126 69 B 37 58 C 15 73 0 A 37 58 B 15 73 C 126 69 1 15 720 R 27 500 126 60

6 Given a 10 b 6 and c 14 use the Law of Cosines to solve the triangle for the value of B Round your answer to two decimal places 0 60 33 0 80 44 38 21 120 00

5 Use the Heron s formula to find the area of the triangle Round your answer upto two decimal pla a 12 42 b 8 56 c 15 15 29 44 12 51

9 Provide an appropriate response What happens if C 90 when the law of cosines is applied in the form C2 A2 B2 2ab cos C

3 Use the law of Cosines to solve the given triangle Round your answer to two decimal places A 130 b 7 c 8 a 13 60 B 25 C 25 a 8 B 23 22 C 26 78 a 7 B 23 22 C 26 78 a 13 60 B 26 78 C 23 22 60 R 22 229 26 790

5 Select a value for b such that a triangle with A 33 and a 4 has no soluti A A 3 67 14 68 1 84 O 1 47 7 24

8 Determine the area of a triangle having the following measurements Round your answer to two decimal places C 73 a 12 5 and b 11 5 55 17 sq units 62 06 sq units 68 96 sq units

1 Use the Law of Sines to solve for b Round your answers to two decimal places A 100 a 75 c 6 b 83 72 b 75 1 72 73

3 Find values for b such that the triangle has no solutions A 24 a 13

4 Convert the expression

6 Use the figure below to determine the exact value of the given function csc 20 2 8

10 Establish the identity algebraically Remember your answer is not a yes or no answer You are to show that one side is equal to the other co csc u cotu csc u cot u

32 8 The range of a projectile fired at an angle with the horizontal and with an initial velocity of vo feet per second is measured in feet A golfer strikes a golf ball at 90 feet per second Ignoring the effects of air resistance at what angle must the golfer hit the ba so that it travels 150 feet Round your answer to the nearest degree 0 36 O 41 14 18 v sin 20 where r is 27

7 Use the figure below to find the exact value of the given trigonometric expression 0 sin 2 24 figura not 10 accari to sagla

3 Use the sum to product formulas to rewrite the sum or difference as a product cos 30 cos 80

5 The mach number M of an airplane is the ratio of its speed to the speed of sound When an airplane travels faster than the speed of sound th sound waves form a cone behind the airplane see figure The mach number is related to the apex angle of the cone by sin 0 5 1 M

2 Use the sum to product formulas to rewrite the sum or difference as a product sin 90 sin 70 O2 sin 90 cos 70 O2 sin 80 sin 0 O 2 cos 80 sin 0 O2 cos 80 cos 0 2 sin 8A COSA

1 Use the product to sum formulas to rewrite the product as a sum or difference sin COS 3 6

6 5 4 3 2 1 7 6 5 4 3 2 41 1 2 3 Slope m 7 6 7 1 Find the slope of the line 2 3 4 5 6

Which of the following points are on the graph of f x 11 Mark all the correct ones 0 1 1 0 O 1 11 1 11 1 0 1 11 1 1 11

Find all solutions of the equation in the interval 0 27 List the solutions in increasing order If there is no solution enter NO SOLUTI 24 sin x 24 12 cos x X X X TU 3 X n 2 50 2 t 4 3 n 2 57 3 n 2 X n 2 X smallest value largest value

Find zw and 2 10 10 W 3 1 Leave your answers in polar form What is the product co Simplify your answers Type exact answers using radicals as needed Use indogers or tractions for any numbers in the expression Type any angle measures in degrees Use angle greater than or equal to 0 and less than 360 What is the quotient con Simplify your answers Type exact answers using radicals as needed Use integers or fractions for any numbers in the expression Type any angle measures in degrees Use angle s greater than or equal to 0 and less than 360

Systems of linear inequalities and Linear programming 12 A factory manufactures two kinds of ice skates racing skates and figure skates The racing skates require 8 work hours in the fabrication department whereas the figure skates require 4 work hours there The racing skates and the figure skates each require 2 work hours in the finishing department The fabricating department has available at most 224 work hours per day and the finishing department has no more than 72 work hours per day available The factory must also produce at least 16 racing skates If the profit on each racing skate is 15 and the profit on each figure skate is 12 how many of each should be manufactured each day to maximize profit Assume that all skates made are sold how the feasible region and the corner points

4 Graph the image of parallelogram CDEF after a dilation with a scale factor of at the origin fa 10 6 4 2 centered

Write an expression for the general or nth term an for the arithmetic sequence 3 8 13 18 The general term of the arithmetic sequence is an Do not factor Use integers or fractions for any numbers in the expression

Use the Law of Cosines to determine the length of side x 37 15 A 17 X C 24 30 30 B

Describe the end behavior of the left side of the function f x x5 3x 2 y approaches infinity as x approaches negative infinity y approaches negative infinity as x approaches infinity Oy approaches negative infinity as x approaches negative infinity O y approaches infinity as x approaches infinity

Use long division or synthetic division to divide the following polynomials 3x x 3x 1 x 3

Which point is not on the Unit Circle O 5 3 2 3 3 5 4 5 O 3 4 7 4 O 2 7 3 7

Find the length of the side labeled x in the following triangle 8 4 62 6 93 X 30 4