math midterm study
⁻⁴/¹⁷
What is the value of n? (81³ⁿ⁺²)/(243⁻ⁿ) = 3⁴
-2
What is the value of n? (¹/₁₂₈) = 2⁵ⁿ⁺³
⁴/₃
What is the value of n? (¹/₂₅)ⁿ⁻² = (125)ⁿ÷³
⁻¹/₆
What is the value of n? (¹/₆)³ⁿ⁺² × 216³ⁿ = (¹/₂₁₆)
-6
What is the value of n? (¹/₇)ⁿ⁺⁴ - 8 = 41
all real numbers
What is the value of n? (¹/₈)²ⁿ = 4⁻³ⁿ
½
What is the value of n? 10⁻³ⁿ × 10ⁿ = ¹/₁₀
-5
What is the value of n? 2⁴ⁿ = 32ⁿ⁺¹
12
What is the value of n? 2⁵ⁿ = 8²ⁿ⁻⁴
3
What is the value of n? 36⁻³ⁿ⁺³ = (¹/₂₁₆)ⁿ⁺¹
³/₂
What is the value of n? 3⁶ⁿ⁻³ = 9²ⁿ
⁻²/₃
What is the value of n? 3⁻²ⁿ⁺¹ × 3⁻²ⁿ⁻³ = 3⁻ⁿ
1
What is the value of n? 4ⁿ = (½)ⁿ⁻³
5
What is the value of n? 4ⁿ⁻² - 22 = 42
¹¹/₂
What is the value of n? 4ⁿ⁻³ = 32
2
What is the value of n? 64²ⁿ⁺⁴ = 16⁵ⁿ
¹/₄
What is the value of n? 9²ⁿ⁺³ = 2187
¹¹/₃
What is the value of x?
Integers
Z = {...-3, -2, -1, 0, 1, 2, 3, ...}
Relation
a set of ordered pairs: { (1,4), ( 1,5), (4,5), (3,2)}
Which set shows all the solutions for ³√64? a. 4 only b. -4 and +4 c. 8 only d. -8 and +8
a.
Which set is the solution for x³ = -343? a. ³√7 b. -7 c. 7 d. ±7
b.
Distance Formula (Using Pythagorean Theorem)
finds the length of line segment
proportional
ratios are equivalent
non-proportional
ratios are not equivalent
non-proportional
the line does not pass through the point of origin in the graph
proportional
the line passes through the point of origin in the graph
(10, -1)
x - y = 11 2x + y = 19
Solve for x. x² = 0
x = 0
Solve for x. x³ = 0
x = 0
Solve for x. x³ = 0.0027
x = 0.03
Solve for x. x³ = 1
x = 1
Solve for x. ³√x = 10
x = 1000
Solve for x. ³√x = 5
x = 125
Solve for x. x³ = 8/27
x = 2/3
Solve for x. √x = 5
x = 25
Solve for x. x² = 0.49
x = ±0.7
Solve for x. x² = 1
x = ±1
Solve for x. x² = 2.25
x = ±1.5
Solve for x. x² = 4/36
x = ±1/3
Solve for x. x² = 225
x = ±15
Solve for x. x² = 256
x = ±16
Solve for x. x² = 16/49
x = ±4/7
infinite solutions
y = -4x + 2 4x + y = 2
no solution
y = 2x + 5 y = 2x - 7
(2, 3)
y = 5x - 7 -3x - 2y = -12
(2, 1)
y = 6x - 11 -2x - 3y = -7
Linear functions
y = Mx + b
Quadratic functions
y = a times an unknown number to the second power + bx + c
Exponential functions
y = a times b to the power of x
(5, 10)
y = x + 5 3x + y = 25
non-proportional
y=1/2x-3
proportional
y=1/3x
proportional
y=12x
proportional
y=4x
non-proportional
y=6x+1
non-proportional
y=9x-2
Natural Numbers
{1,2,3...}
22 / 10
2.2
Put in scientific notation: 300,000
3 x 10⁵
Inputs (Domain)
The set of x-values
Put in standard notation: 1.50 x 10⁻⁴
0.000150
4.6 / 1,000
0.0046
Put in standard notation: 6.330 x 10⁻³
0.006330
0.075 / 10
0.0075
0.04 x 10
0.4
43 x 0.01
0.43
0.05 x 10
0.5
941 x 0.001
0.941
³√1
1
1,139.4 / 1,000
1.1394
199 / 100
1.99
proportional
1/11 and 3/33
proportional
1/2 and 4/8
proportional
1/3 and 2/6
proportional
1/5 and 3/15
√121
11
√144
12
12.45 / 1
12.45
1,298 / 10
129.8
√225
15
1,563 / 100
15.63
1.58 x 100
158
√289
17
1.98 x 10
19.8
x = 5
2 (x + 1) = 3x - 3
Put in proper scientific notation: 235 x 10^-4
2.35 x 10^-2
Put in proper scientific notation: 2350 x 10^0
2.35 x 10^3
Put in proper scientific notation: 23.5 x 10^5
2.35 x 10^6
Put in scientific notation: 24,327
2.4327 x 10⁴
Put in scientific notation: 24,327,000
2.4327 x 10⁷
2.91 x 1
2.91
non-proportional
2/10 and 5/20
25.3 x 100
2530
2.95 x 1000
2950
infinite solutions
2x + 8y = 6 -5x - 20y = -15
(4, 3)
2x - 3y = -1 y = x - 1
0.03 x 100
3
Put in scientific notation: 30
3 x 10¹
Put in scientific notation: 3
3 x 10⁰
34 / 10
3.4
341 / 100
3.41
non-proportional
3/4 and 1/10
proportional
3/4 and 6/8
non-proportional
3/9 and 4/16
Put in standard notation: 3 x 10²
300
Put in standard notation: 3 x 10⁵
300,000
329.4 / 10
32.94
Put in standard notation: 3.25 x 10²
325
100 x 3.49
349
Put in proper scientific notation: 400 x 10^-8
4 x 10^-6
Put in proper scientific notation: 0.004 x 10^5
4 x 10^2
Put in proper scientific notation: 40 x 10^6
4 x 10^7
non-proportional
4/12 and 1/2
non-proportional
4/20 and 4/16
³√125
5
Put in proper scientific notation: 0.005 x 10^3
5 x 10^0
Put in proper scientific notation: 500 x 10^-2
5 x 10^0
Put in proper scientific notation: 50 x 10
5 x 10^2
Put in standard notation: 5.72×10⁹
5,720,000,000
5,125 / 1,000
5.125
Put in scientific notation: 0.053
5.3 x 10⁻²
Put in scientific notation: 0.0053
5.3 x 10⁻³
53.7 / 10
5.37
0.001 x 5600
5.6
Put in scientific notation: 572
5.72×10²
Put in scientific notation: 57,200
5.72×10⁴
Put in standard notation: 5.72×10⁴
57,200
592.3 x 0.1
59.23
(-6, 0)
5x + 4y = -30 3x - 9y = -18
621 / 100
6.21
Put in scientific notation: 0.0078
7.8 x 10⁻³
³√512
8
√64
8
83.2 / 10
8.32
894.1 x 0.01
8.941
(-1, -8)
8x + y = -16 -3x + y = -5
Linear
A function that forms a straight line when graphed
Function f(x)
A relation for which each element of the domain corresponds to exactly one element of the range
Function
A relationship that assigns exactly one output value for each input value
Relation
A set of ordered pairs
Vertical Line Test
A test use to determine if a relation is a function. A relation is a function if there are no vertical lines that intersect the graph at more than one point.
dependent variable
A variable (y) that depends on one or more other variables
Standard Form of a Linear Equation
Ax+By=C, where A, B, and C are integers and A is not negative
Negative Slope
Falls from left to right
True or False? √1 is not equal to ³√1.
False. Both statements are equivalent to 1.
Parallel Lines
Have equal slopes (gradients)
Perpendicular Lines
Have opposite reciprocal slopes (gradients)
Nonlinear
Not a straight line when graphed
Rational Numbers
Q : Numbers that can be written as the ratio of two integers; includes decimals that terminate or repeat
Irrational Numbers
Q' ( or I ) : Any number that is not a rational number, includes decimals that neither terminate nor repeat
Positive Slope
Rises from left to right
(⁵/₁₁)x²y⁷
Select the equivalent expression. (10x⁴y⁸)/(22x²y)
32x⁴y⁵
Select the equivalent expression. (2xy)³×(4xy²)
-96x⁷y⁶
Select the equivalent expression. (2xy)⁵(-3x²y)
8x⁴y⁴z⁴
Select the equivalent expression. (2xy²z²)×(4x³y²z²)
16x⁸
Select the equivalent expression. (2x²)⁴
144x¹⁸
Select the equivalent expression. (2x²)⁴×(3x⁵)²
8x⁶y⁹
Select the equivalent expression. (2x²y³)³
z¹⁰/(4x⁴y⁶)
Select the equivalent expression. (2x³y⁻⁵z⁷)/(8x⁷yz⁻³)
(⁻⁴/₅)x⁴
Select the equivalent expression. (2x⁴y)³/(-10x⁸y³)
6x¹⁵y¹⁰
Select the equivalent expression. (3x¹⁰y⁸)×(2x⁵y²)
9x⁸y¹⁶
Select the equivalent expression. (3x⁴y⁸)²
y⁶/(27x¹²)
Select the equivalent expression. (3x⁴y⁻²)⁻³
-y¹⁰/(4x²z)
Select the equivalent expression. (4x³y¹⁰z⁻⁴)/(-16x⁵z⁻³)
8x⁷/y²
Select the equivalent expression. (4x⁴y³)×(2x³y⁻⁵)
2/(5x⁹y⁴)
Select the equivalent expression. (4x⁻⁵y⁻³)(2x²) / (20x⁶y)
15x⁵y³
Select the equivalent expression. (5x³y⁵)×(3x²y⁻²)
4x²
Select the equivalent expression. (8x⁵)/(2x³)
x¹⁰y¹⁵
Select the equivalent expression. (x²y³)⁵
y³/x
Select the equivalent expression. (x³y⁵)/(x⁴y²)
x⁷y¹⁵
Select the equivalent expression. (x³y⁶)×(x⁴y⁹)
1/y³
Select the equivalent expression. (x⁴y)(xy⁷)/(x⁵y¹¹)
x²y³
Select the equivalent expression. (x⁴y⁹)/(x²y⁶)
x²⁴
Select the equivalent expression. (x⁶)⁴
x⁸⁰
Select the equivalent expression. (x⁸)¹⁰
y³/x³
Select the equivalent expression. (y²x)×(yx⁻⁴)
-6x⁸y¹²
Select the equivalent expression. -6(x²y³)⁴
x²²
Select the equivalent expression. x¹⁰ × x¹²
x¹³
Select the equivalent expression. x⁵ × x⁸
x
Select the equivalent expression. x⁵/x⁴
x⁵/y⁴
Select the equivalent expression. x⁵y⁻⁴
x³
Select the equivalent expression. x⁹/x⁶
1/x³
Select the equivalent expression. x⁻³
What is the principal square root of 36 and why?
Six is the principal square root because it is the positive root.
x = 9.6
Solve 0.3x - 4 = 0.8 - 0.2x
x = 0.4
Solve 15 - 3x = x + 13 + x
x = 4
Solve 3 (x +1) = 2x + 7
x = 0.5
Solve 3 - 2(x - 1) = 2 + 4x
x = 1/3
Solve 3(2x + 2) - 4 = 3(x + 1)
x = 4/3
Solve 4(2x - 1) + 5 = 5(x + 1)
x = 1
Solve 4x + 2 = 3 (6 - 4x)
x = 12
Solve 4x - 5 = 3x + 7
x = -8
Solve 6x - 4 (2x + 1) = 12
x = 3
Solve 7x + 3 - 2x = 4x + 6
x = -4
Solve 8x - 3 = 5 (2x + 1)
x = 1.5
Solve 8x - 5 = 4x + 4 - 2x
Parabola
The U shaped curve of a quadratic function and each time you have x squared
Domain
The possible values for the input, or independent variable of the function, the x value
Range
The possible values for the output, or the dependent variable of the function, the y value
Outputs (Range)
The set of y-values
Zero Slope
The slope of a horizontal line m=0
Undefined Slope
The slope of a vertical line
Independent variable
The variable that represents the input and causes the change in the dependent variable; x-axis label
73 x 0.01
0.73
√1
1
Slope-Intercept Form : Gradient-intercept form
(in USA ) y =mx +b (in IB): y =mx +c
³√-1000
-10
no solution
-3x + 3y = 4 -x + y = 3
(6, -6)
-4x - 2y = -12 4x + 8y = -24
³√-8
-2
³√-729
-9