Write Expression/Equation from the Important Info

M8A1.a – Represent a given situation using algebraic expressions or equations in 1 variable.

-underline important words and info

-write expression/equation from the important info

Ex. 1  2 - x ∙ 2

two less thandouble a number (switch ‘em)

Answer: 2x – 2

Ex. 2  2 ∙ (y + 5)

two times the sum of y and five

Answer: 2(y+5)

Ex. 3  5 - x

five decreased by a number

Answer: 5 – x

Ex. 4  ∙2 x + 4

double a number increased by four

Answer: 2x + 4

M8A1.b – Simplify and evaluate algebraic expressions.

-Combine like terms (to simplify)

-Replace the variable with its value and solve (to evaluate)

Ex. 1  4x + 3(x + 1)

4x + 3x + 3

Answer: 7x + 3

Ex. 2  5x – 2y + 3x + 6y – 1

8x + 4y - 1

Answer: 8x + 4y – 1

Ex. 3  3t ÷ 3 + t when t = 13

3(13)÷ 3 + 13

39 ÷ 3 + 13

13 + 13

Answer: 26

Ex. 4  x + x2 + x ÷ 3 when x = 3

3 + 32 + 3 ÷ 3

3 + 9 + 3 ÷ 3

3 + 9 + 1

12 + 1

Answer: 13

M8A1.c – Solve algebraic equations in one variable, including those involving absolute value.

-1-step equations

Find the # getting in the way of the variable.
What operation is happening?
Perform the inverse operation to both sides of the equal sign.

Ex. 1  3x = 15

3 3

Answer: x = 5

Ex. 2  x – 4 = 6

+4 +4

Answer: x = 10

Ex. 3  (x/2)2 = (-7)2

Answer: x = -14

Ex. 4  x + 8 = -11

-8 -8

Answer: x = -19

M8A1.c

-2-step Equations

Find the #s getting in the way of the variable
Determine which is furthest
Perform the inverse operation to both sides
Determine # still getting in the way
Perform the inverse operation to both sides

Ex. 1  2x + 3 = 13

-3 -3

2x = 10

2 2

Answer: x = 5

Ex. 2  x/4 – 2 = 7

+2 +2

(x/4)4 = (9)4

Answer: x = 36

M8A1.c

-Absolute value equations

Get the absolute value symbol by itself
Set up 2 equations (1 = positive, 1 = negative)
Determine if it’s 1-step or 2-step
Follow those steps

Ex. 1  |x – 4| = 12

x – 4 = 12x – 4 = -12

+4 +4 +4 +4

x = 16 x = -8

Answer: x = 16 and -8

Ex. 2  |3x – 6| = 15

3x – 6 = 153x – 6 = -15

+6 +6 +6 +6

3x = 21 3x = -9

3 3 3 3

x = 7 x = -3

Answer: x = 7 and -3

Ex. 3  |5x| + 10 = 30

-10 -10

|5x| = 20

5x = 205x = -20

5 5 5 5

x = 4 x = -4

Answer: x = 4 and -4

M8A1.c

-Distributive Property Equations

Distribute # outside to everything inside
Determine 1-step or 2-step
Follow those steps

Ex. 1  -2(x – 4) = 12

-2x + 8 = 12

-8 -8

-2x = 4

-2 -2

Answer: x = -2

Ex. 2  5(x + 5) = 40

5x + 25 = 40

-25 -25

5x = 15

5 5

Answer: x = 3

M8A1.c

-Equations with variables on both sides

Locate the smallest variable
Perform the inverse operation to both sides
Determine 1-step or 2-step
Follow those steps

Ex. 1  5x – 6 = 2x + 9

-2x -2x

3x – 6 = 9

+6 +6

3x = 15

3 3

Answer: x = 5

Ex. 2  x – 12 = 2x - 5

-x -x

-12 = x – 5

+5 +5

-7 = x

Answer: x = -7

M8A1.d – Solve equations involving several variables for one variable in terms of the others.

-Circle the variable you’re trying to get alone.

-List everything getting in the way.

-List the operations occurring.

-Perform the inverse operations, beginning with those that are furthest away from the circled variable.

Ex. 1  y = m x + b for mx, b

-b -bmultiplication, addition

y – b = m x

x x

Answer: y – b = m

Ex. 2  x/ 2 – 4a = 3y for x2, 4a

+4a +4adivision, subtraction

(x/ 2)2 = (3y + 4a)2

x = 2(3y + 4a)

Answer: x = 6y + 8a

M8A2 – Understand and graph inequalities in one variable.

-Solving Inequalities

Determine the type of inequality.
Follow the same steps as equations.
Don’t forget – if you multiply or divide by a negative, flip the inequality symbol.

-Graphing Inequalities on # line

< and > open circle
≤ and ≥ closed circle
< and ≤ shade left
> and ≥ shade right

Ex. 1  x + 5 > 12

- 5 - 5

Answer: x > 7

6 7 8

Ex. 2  -4x≥20

-4 -4

Answer: x ≤ -5

-6 -5 -4

M8A2

-Absolute value Inequalities

Get absolute value by itself.
Set up 2 inequalities (1 to positive, 1 to negative – don’t forget to flip inequality symbol with the negative)
Solve using same steps as equations.
Don’t forget – if you multiply or divide by a negative, flip the inequality symbol.

-Graphing Absolute Value Inequalities on # line

< and > open circle
≤ and ≥ closed circle
< and ≤ shade inside
> and ≥ shade outside

Ex. 1  |2x – 3| ≥ 13

2x – 3 ≥ 132x – 3 ≤ -13

+3 +3 +3 +3

2x≥16 2x≤-10

2 2 2 2

x ≥ 8 x ≤ -5

Answer:-5 ≥ x ≥ 8

-6 -5 -4-3-2-10 1 2 3 4 5 6 7 8 9

Ex. 2  3|-3x|≤27

3 3

|-3x| ≤ 9

-3x≤ 9-3x ≥ -9

-3 -3 -3 -3

x ≥ -3 x ≤ 3

Answer:-3 ≤ x ≤ 3

-4 -3 -2 -1 0 1 2 3 4

Symbols:

is less than

is greater than

≤is less than or equal to

≥is greater than or equal to

=is equal to

≈is approximately equal to

≠is not equal to

±positive/negative

+add (or positive)

-subtract (or negative)

∙multiply

( )

Vocabulary:

-expression – a mathematical phrase that contains operations, #s, and/or variables

-variable – a letter used to represent a value that can change

-integers – a set of whole #s and their opposites (…-3, -2, -1, 0, 1, 2, 3,…)

-opposites – 2 #s that are an equal distance from 0 on a # line (1 and -1, 5 and -5, etc.)

-evaluate – to find the value of a numerical or algebraic expression

-solution – any value that makes a statement true

Vocabulary:

-equation – a mathematical sentence that shows that 2 expressions are equivalent

-equivalent – having the same value

-inverse operation – operations that undo or cancel out each other

(addition  subtraction, multiplication  division, square root  squaring)

-absolute value – the distance of a # from 0 on # line

-distributive property – multiply the # outside the parentheses to everything inside

-inequality – a mathematical sentence that shows 2 quantities are not equal

M8A2.a,b – Recognize a relation and a function as a correspondence between inputs and outputs, where (in functions) the output for each input must be unique.

-Functions – each input has 1 output (x values should not repeat)

x / y
-2 / 5
-1 / 7
0 / 9
1 / 11

Ex. 1 

Function because each input (x) has 1 output (y).

x / y
1 / -3
0 / 5
2 / 7
1 / -4

Ex. 2 Relation because 1 has two outputs (-3 and -4).

M8A2.a,b

-Functions – each input matches with only 1 output

Ex. 1 

32 Function because each input matches with one output.

4-1

Ex. 2 

02 Relation because 3 matches with two outputs (2 and 5).

Ex. 3 

6 -3 Function because each input matches with only one output.

1 2

M8A3.a,b

-Functions – the vertical line test passes through the graph only once.

Ex. 1 

Relation because the vertical line passes 3 times

Ex. 2 

Function because the vertical line passes 1 time

Ex. 3 

Relation because the vertical line passes 2 times

Ex. 4 

Function because the vertical line passes 1 time

M8A3.e,f – Understand and recognize arithmetic sequences as linear functions with whole # input values.

-identify the common difference

-write the arithmetic formula

-replace the appropriate variables with their values

an = a1 + (n – 1)d

Ex. 1 3, 9, 15, 21, 27,…

+6 +6 +6 +6

an = a1 + (n – 1)d

an = 3 + (n – 1)6

Ex. 2 -7, -9, -11, -13,…

-2 -2 -2

an = a1 + (n – 1)d

an = -7 + (n – 1)(-2)

M8A3.e,f

-Use the formula to find the specified terms.

Arithmetic Formula:

an = a1 + (n – 1)d

Ex. 1 an = 3 + (n – 1)6

a26 = ?

a26 = 3 + (26 – 1)6

a26 = 3 + (25)6

a26 = 3 + 150

a26 = 153

Ex. 2 an = -7 + (n – 1)(-2)

a31 = ?

a31 = -7 + (31 – 1)(-2)

a31 = -7 + (30)(-2)

a31 = -7 – 60

a31 = -67

M8A3.e,f

-Use the recursive formula to find the specified term (previous term must be given).

Recursive Formula:

tn = tn-1 + d

Ex. 1 t18 = 28

tn = tn-1 – 5

t19 = ?

t19 = t19-1– 5

t19 = t18 – 5

t19 = 28 – 5

t19 = 23

Ex. 2 t51 = -15

tn = tn-1 + 3

t52 = ?

t52 = t52-1+ 3

t52 = t51 + 3

t52 = 15

t52 = -12

Vocabulary:

function – an input-output relationship that has exactly 1 output for each input

relation – any relationship between inputs and outputs

vertical line test – draw a vertical line through a graph – if at any place on the graph, it intersects more than once, it is a relation (if the vertical line intersects only once all the way through, it is a function)

arithmetic sequence – a sequence that has a common difference (you add or subtract the same # every time)

Vocabulary:

sequence – an ordered list of numbers

term – an element or number in a sequence

recursive sequence – each term of the sequence is defined as a function of the previous term

linear function – a function whose graph is a straight line

nonlinear function – a function whose graph does not form a straight line

M8A4.b – Determine the meaning of slope and y-intercept in a given situation.

Slope = rise

run

M8A4.a - Interpret slope as a rate of change.

-Find 2 points on the line.

-Count the rise.

-Count the run.

-Write slope as rise

run

Ex. 1 

slope = -2 = -2

Ex. 2 

slope = 10 = 2

M8A4.a

Finding slope using the slope formula:

- Label the ordered pairs

- Write the slope formula

- Bring over the ____-____

- Replace the variables with their values

- Solve (don’t forget your integer rules)

Ex. 1  x1 y1 x2 y2

(-3, 4) (2, -1)

m = y2 – y1 = -1 – 4 = -5 = -1

x2 – x1 2 – (-3) 5

Ex. 2 x1 y1 x2 y2

(0,-1) (-5,0)

m = y2 – y1 = 0 – (-1) = 1 = -1

x2 – x1 -5 – 0 -5 5

***Reminders about graphing ordered pairs***

When graphing ordered pairs:

-always begin at the origin

-an ordered pair is (x,y)

-if x is +, go right; if x is –, go left

-if y is +, go up; if y is –, go down

-graph the x before the y

Ex. 1 (-3,5)

from the origin, go left 3 and up 5, put a point

Ex. 2 (4, -1)

from the origin, go right 4 and down 1, put a point

Ex. 3 (2,0)

from the origin, go right 2, put a point

Ex. 4 (0,-4)

from the origin, go down 4, put a point

M8A1.d

-Writing equations in slope-intercept form:

circle the y
list everything getting in the way
list the operations
perform the inverse operations
write in slope-intercept form (y=mx+b)
identify slope and y-intercept

Ex. 1 2y – 10x = 82, -10x

+ 10x +10xx, -

2y = 10x + 8

2 2 2

y = 5x + 4slope = 5y-int. 4

Ex. 2 -3x – 2y = 12-3x, -2

+3x +3xx, -

-2y = 3x + 12

-2 -2 -2

-3

y = 2 x – 6 slope = -3/2y-int. = -6

M8A4.c – Graph equations in the form of y=mx+b.

-Be sure equation is in the form of y=mx+b.

-Identify slope (m) and y-intercept (b)

-Graph y-intercept

-Graph slope (rise & run)

Ex. 1 

y = -2x + 1

m = -2/1

b = 1

Ex. 2 

y = 2x + 3

m = 2/1

b = 3

M8A4.e – Graph the solution set of an inequality, identifying whether the solution set is an open or closed half plane.

-Graphing inequalities on coordinate plane:

inequality should be in slope-intercept form
identify and graph slope and y-int.
< and > open line
≤ and ≥ closed line
< and ≤ shade left or below
> and ≥ shade right or above
check shading with a point – preferably (0,0)

Ex. 1 y > -2x + 3

y > -2x + 3 (0,0)

0 > -2(0) + 3

0 > 0 + 3

0 > 3

Ex. 2 y ≥ 2x – 5

y ≥ 2x – 5 (0,0)

0 ≥ 2(0) – 5

0 ≥ 0 – 5

0 ≥ -5

Vocabulary:

-slope – the steepness of a line; the constant rate of change; represented by m in y=mx+b; rise

run

-y-intercept – where the graph crosses the y-axis; the starting point; represented by b in y=mx + b

-constant rate of change – when you increase or decrease by the same amount to get to the next term to get to the next term or point (common difference)

-ordered pairs – a pair of #s that can be used to locate a point on the coordinate plane (x,y)

- slope formula – used to find the slope from 2 given ordered pairsm = y2 – y1

x2 – x1

- slope-intercept form = a linear equation written in the form of y=mx + b, where m represents slopeand b represents y-int.

- coordinate plane – a plane formed by the intersection of a horizontal # line (x-axis) and a vertical # line (y-axis)

- plot – to place points on the coordinate plane based on the ordered pair (x,y) see page 28

M8A4.f – Determine the equation of a line given a graph, numerical info that defines the line, or a context involving a linear relationship.

-Writing Equation of a Line from Sequences:

identify 1st term (y-int.)
identify common difference (slope)
replace m & b in y=mx+b

Ex. 1 5, 9, 13, 17, 21,…

+4 +4 +4 +4

m = 4b = 5

y = mx + b

y = 4x + 5

Ex. 2 -2, -5, -8, -11,…

-3 -3 -3

m = -3b = -2

y = mx + b

y = -3x – 2

M8A4.f

-Writing Equation of a Line from Tables:

be sure the input values (x) are consecutive
find the common difference between the output values (y); this will be the slope (m)
find the y value when x = 0; the will be the y-int. (b)
replace m & b in y=mx+b

Ex. 1 x y

110 y = mx =b

214 y = 4x + 6

m = 4b = 6

Ex. 2 x yx y

-1 -8 -1 -8

1-2 0 -5y = mx =b

0 -51 -2y = 3x – 5

3 4 2 1

3 4

m = 3b = -5

M8A4.f

-Writing Equation of a Line from Graphs:

identify the y-int. (b) where the graph crosses the y-axis
identify the rise and run and write as slope (m)
replace m & b in y=mx+b

Ex. 1 

Ex. 2 

M8A4.f

-Writing Equation of a Line from Ordered Pairs:

put ordered pairs into a table
follow steps on page 35

Ex. 1 (0,5), (1,3), (2,1), (3,-1)

x y

53 y = mx =b

7-1 y = -2x + 5

m = -2b = 5

Ex. 2 (2,10), (-1,-8), (0,-2), (1,4)

x y

-1 -8

1 -2y = mx =b

2 4y = 6x – 2

3 10

m = 6b = -2

M8A4.f

-Writing Equation of a Line from Word Problems:

find the # being added once – this will be the y-int. (b)
find the # that will be multiplied by the input values – this will be the slope (m)
replace m & b in y=mx+b

Ex. 1  A cab fare costs $3.00 for a base fee. Also, you

must pay $0.50 per mile. Write the equation of the line from the given information.

$3 base fee added once (y-int.)

$0.50 per mile (slope)

y=mx+b

Answer:y=0.50x + 3

M8A4.f

Ex. 2 A cell phone plan costs $5.00 for the first 200 texts. Also, you must pay

$0.20 for each additional text. Write an equation for this information.

$5 for 200 texts (y-intercept)

$0.20 for each additional text (slope)

y=mx + b

y = 0.20x + 5

Ex. 3 Sara has $35 in savings. She earns $40 per week at her part-time job.

Write an equation to show how much money she will have over time.

$35 saving (y-intercept)

$40 per week (slope)

y=mx + b

y = 40x + 35

M8A4.f

-Writing Equations of a Line from Lines of Best Fit (Scatter Plot):

determine the line of best fit (points should be divided in half if possible, and line should follow the slope trend)
write an equation from your line of best fit and choose the best estimate

Ex. 1 

Answer: y = -x + 14

Ex. 2 

Answer: y = 0.25x + 2

M8A5.b – Solve systems of equations graphically and algebraically.

-Solving Systems of Equations through Elimination:

solve 1 equation for either x or y
replace that variable with its value in the other equation and solve the equation for the remaining variable
replace its value into the other equation and solve
check the solution in an original equation

Ex. 1 2x + 4 = 10

x – 3y = 10

Step 1: x – 3y = 10Step 2: 2x + 4y = 10

+3y +3y 2(3y+10) + 4y = 10

x = 3y + 10 6y + 20 + 4y = 10

10y + 20 = 10

- 20 -20

Step 3: x – 3y = 10 10y = -10

x – 3(-1) = 10 10 10

x + 3 = 10y = -1

-3 -3

x = 7

Step 4: 2x + 4y = 10

2(7) + 4(-1) = 10

14 – 4 = 10

10 = 10

M8A5.b

Ex. 2 2x – 3y = -2

4x + y = 24

Step 1: 4x + y = 24Step 2: 2x – 3y = -2

-4x -4x 2x – 3(-4x + 24) = -2

y = -4x + 24 2x + 12x – 72 = -2

14x – 72 = -2

+ 72 +72

Step 3: 4x + y = 24 14x = 70

4(5) + y = 24 14 14

20 + y = 24 x = 5

-20 -20

y = 4

Step 4: 2x – 3y = -2

2(5) – 3(4) = -2

10 – 12 = -2

-2 = -2

Ex. 3 2x + y = 9

3x – y = 16

Step 1: 2x + y = 9Step 2: 3x – y = 16

-2x -2x 3x – 1(-2x + 9) = 16

y = -2x + 9 3x + 2x – 9 = 16

5x – 9 = 16

+ 9 +9

Step 3: 2x + y = 9 5x = 25

2(5) + y = 9 5 5

10 + y = 9 x = 5

-10 -10

y = -1

Step 4: 3x – y = 16

3(5) – (-1) = 16

15 + 1 = 16

16 = 16

M8A5.b

-Solving Systems of Equations through Elimination/Combination:

determine which variable is easiest to cancel out
multiply an equation by the inverse, if necessary
cancel out the variable and combine like terms to solve for the remaining variable
replace its value into 1 equation to solve for the other variable
check the solution in the other equation

Ex. 1 2x + y = 92x + y = 9

3x – y = 162(5) + y = 9

5x = 2510 + y = 9

5 5-10 -10

x = 5 y = -1

3x – y = 16

3(5) – (-1) = 16

15 + 1 = 16

16 = 16

M8A5.b

Ex. 2 (2x – 3y = -2)-2-4x + 6y = 4

4x + y = 244x + y = 24

7y = 28

y = 4

2x – 3y = -2

2x – 3(4) = -2

2x – 12 = -24x + y = 24Answer: (5,4)

+12 +124(5) + 4 = 24

2x = 10 20 + 4 = 24

2 2 24 = 24

x = 5

Ex. 2 2x + 4y = 10-2x + 6y = -20

(x – 3y = 10)-22x + 4y = 10

10y = -10

10 10

y = -1

2x + 4y = 10

2x + 4(-1) = 10

2x – 4 = 10x – 3y = 10Answer: (7,-1)

+4 +47 – 3(-1) = 10

2x = 14 7 + 3 = 10

2 2 10 = 10

x = 7

M8A5.b

-Solving Systems of Equations from Graphs:

find the intersection point of the 2 lines on the graph

Ex. 1 

y = x + 2

y = -x + 4

Answer: (1,3)

Ex. 2 

y = -3x + 6

y = 2x – 4

Answer: (2,0)

M8A5.b

-Writing Systems of Equations from Word Problems

Ex. 1 Together Mike and Sara swam 27 laps in a pool. Sara swam twice as

many as Mike did. Write a system of equations and solve it to see how many laps each

swam.

Mike & Sara swam 27

Sara swam twice as many as Mike

m + s = 27m + 2m = 272m = s

2m = s 3m = 272(9) = s

3 318 = s

m = 9

M8A5.b

-Writing Systems of Inequalities from Word Problems

Ex. 1 Suppose you can work a total of no more than 10 hours per week at your 2 jobs. Babysitting pays $5 per hour and your cashier job pays $10 per hour. You need to earn at least $60 per week to pay for your car. Write the system of inequalities to match this information.

Work no more than 10 hours/week

Babysitting = $5/hour

Cashier = $10/hour

Need to earn at least $60/week

x + y ≤ 10

5x + 10y ≥ 60

M8A5.b

-Solving Systems of Inequalities from Graphs

graph each inequality
determine the common shading (each point in this shaded area represents the solution set)

Ex. 1 

solution set (the purple area where the pink shading overlaps the blue shading)

M8A5.b

Ex. 2 

solution set (the double shaded area where the pink shading overlaps the blue shading)

Ex. 3 

solution set (the double shaded area in

orange where the purple shading overlaps the blue shading)

Vocabulary:

-linear equation – (equation of a line) an equation that forms a straight line when graphed

-consecutive – in order as they would be on a number line (…-2, -1, 0, 1, 2,…)

-scatter plot – a graph with points plotted to show a possible relationship between 2 sets of data

-intersection point – the point on the graph where the lines cross one another

Vocabulary:

-system of equations – a set of 2 or more equations that contain 2 or more variables

-system of inequalities – a set of 2 or more inequalities that contain 2 or more variables and have an entire solution set

-solution set – the set of values that make a statement true

-line of best fit – a straight line that comes closest to the points on a scatter plot

M8N1.d – Understand that the square root of 0 is 0 & that every positive # has 2 square roots that are opposite in sign.

Ex. 1 √0 = 0 (because 0 x 0 = 0)

Ex. 2 √25 = 5 and -5 (because 5 x 5 = 25 and -5 x -5 = 25)

Answer can be written as √25 = ±5

Ex. 3 √121 = 11 and -11 (because 11 x 11 = 121 and -11 x -11 = 121)

Answer can be written as √121 = ±11

M8N1.f – Estimate square roots of positive #s.

-Find the perfect squares closest to the given # and estimate the closest square root.

14 5

Ex. 1 √95 ≈ ____√81√95√100√95 ≈ _10 or 9.7_

9 10

Ex. 2 √11 ≈ ____√9√11√16√11 ≈ _3 or 3.1_

3 4

M8N1.c – Recognize square roots as points and lengths on a # line.

-Follow the steps to estimate the square root (if needed) & locate the point(s) on the # line.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Ex. 1 √14

√9√14√16

3 4

3 √14 4

Ex. 2√50

√49√50√64

7 8

7 √50 8

M8N1.g – Simplify, add, subtract, multiply, and divide expressions containing square roots.

-If a # isn’t a perfect square, simplify it by:

listing all of the factor pairs
choosing the one with the largest perfect square root
taking the square root of that # and using it as your coefficient
leaving the other # in square root

Ex. 1 √5001 x 500*5 x 100*

√100 x √52 x 25010 x 50

10√54 x 125

Ex. 2 √751 x 75

√25 x √33 x 25

5√35 x 15

M8N1.g

Ex. 3 6 √321 x 32

*2 x 16*

√16 x √24 x 8

6 x 4 √2

Answer:24√2

Ex. 4 4 √501 x 50

*2 x 25*

√25 x √25 x 10

4 x 5 √2

Answer:20√2

Ex. 5 10 √271 x 27

*3 x 9*

√9 x √3

10 x 3 √3

Answer:30√3

Ex. 6 5 √36

5 x 6

Answer: 30

M8N1.g

To add or subtract square roots, you must have like terms. Then, just add or subtract the coefficients.

Ex. 15√2 – 12√2

Answer: -7√2

Ex. 23√5 + 7√5

Answer: 10√5

Ex. 35√2 + 3√18

√9 x √2

5√2 + 3 x 3√2

5√2 + 9√2

Answer: 14√2

M8N1.g

-To multiply square roots, you:

multiply the coefficients
multiply the square roots
simplify the square roots (if necessary)

Ex. 13√2 x 5√8

15 √16

15 x 4

Answer: 60

Ex. 24√5 x 2√15

8 √75

√25 x √3

8 x 5√3

Answer: 40√3

M8N1.g

-To divide square roots:

simplify the fraction (if possible)
you can’t leave a radical in the denominator, so use that radical to multiply by the numerator & denominator
simplify the fraction again

Ex. 13√5

√2

3√5 x √2 = 3√10 = 3√10

√2 √2 √4 2

Ex. 26√5

3√3

6√5 = 2√5 x √3 = 2√15 = 2√15

3√3 √3 √3 √9 3

M8N1.i – Simplify expressions containing integer exponents.

-Use the base as a factor however many times the exponent specifies.

Ex. 145(4 is the base and will be used as a factor 5 times)

4 x 4 x 4 x 4 x 4

16 x 4 x 4 x 4

64 x 4 x 4

256 x 4

Answer: 1,024

Ex. 2 (-3)4

-3 x -3 x -3 x -3

9 x -3 x -3

-27 x -3

Answer: 81

Ex. 3 -34

-(34)

-(81)

Answer: -81

M8N1.i

-When multiplying powers with the same base, add the exponents.

Ex. 1y 5 x y 3

y5+3

Ex. 2 3 x 35 x 32

31 x 35 x 32

3 1 + 5 + 2

3 8

Ex. 3 53 x 54 x 62

5 3 + 4 x 62(you can only add the exponents if the bases are the same)

57 x 62

M8N1.i

-When dividing powers with the same base, subtract the exponents.

Ex. 1y7

y7-4

Answer:y3

Ex. 2 48

48-1

Answer:47

Ex. 3 2a10b5

6a7b6

1a 10-7

3b 6-5

1a3

3b1

Answer:a3

Ex. 4 -10xy4

5x3y7

-2 .

1x3-1y7-4

-2 .

x2y3

M8N1.i

-When raising a power to a power, multiply the exponents.

Ex. 1(x 5)3

x 5 x 3

x 15

Ex. 2 (5-3)-4

5-3 x -4

5 12

Ex. 3 (105)3

10 5 x 3

1015

M8N1.i

-Remember that you don’t want to keep negative exponents.

To get rid of negative exponents, move it across the fraction bar.

Ex. 12a -4b-3

2 .

a4b3

Ex. 2 3x2y-3

4z-4

3x2z4

4y3

Ex. 3 2a-3b-2

6c-1d-2

1c1d2

3a3b2

cd2

3a3b2

-Remember that anything to the 0 power equals 1.

Ex. 1 50 = 1Ex. 3 (-4)0 = 1

Ex. 2 280 = 1Ex. 4 -80 = -(80) = -1

M8N1.j – Express and use numbers in scientific notation.

-To write numbers from scientific notation to standard form:

Use the coefficients and move the decimal the amount of times specified by the exponent on the power of 10

Ex. 15.5 x 103

5.500positive exponent, move decimal to the right

5,000

Ex. 12.5 x 10-2

0.2 5negative exponent, move decimal to the left

0.025

Ex. 3 7 x 105

7 0 00 0 0decimals are at the end of whole #s

700,000

M8N1.j

-To write numbers from standard form to scientific notation:

Move the decimal so that it lands after the first non-zero #
Count the # of times you moved it, this will be your exponent in the power of 10
Large #s  positive exponents

Small #s  negative exponents

Ex. 10.000004224

0.000004224

4.224 x 10-6

Ex. 25,600,000

5,6 00,0 0 0

5.6 x 106

Ex. 3 0.0078

0.0 0 7 8

7.8 x 10-3

M8N1.j

-When multiplying scientific notation problems:

Multiply the coefficients
Use the exponent rules to multiply powers with the same base (add the exponents)

Ex. 1(6 x 10-3) x (7 x 10-6)

42 x 10-9

0.0 00 0 0 0 04 2

0.000000042

Ex. 2(3 x 104) x (2 x 103)

6 x 107

6 0 0 00 0 00

60,000,000

Ex. 3 (-4 x 105) x (5 x 10-3)

-20 x 102

- 2 0

-0.2

Vocabulary

-additive inverse – the sum of a # and its additive inverse is 0 (also called the opposite) 5 and -5, -7 and 7, etc.

-exponent – the number of times a base is used as a factor

-radical – a symbol that is used to indicate square roots √

-significant digits – a way of describing how precisely a # is written

-Square root – one of two equal factors of a nonnegative # (for example 5 is a square root of 25 because 5 x 5 = 25 and -5 is a square root of 25 because -5 x -5 = 25)

-Scientific notation – a representation of real #s as the product of a # between 1 and 10, used for very large #s and small #s

-Rational – a # that can be written as the ratio of 2 integers with a nonzero denominator

-Irrational – a real # whose decimal form is non-terminating add non-repeating that cannot be written as the ratio of 2 integers

Rational #s can be written as:

-fraction

-perfect square roots

-terminating decimals

-repeating decimals

-decimal with a pattern

Examples:

215

346

√16√81√4

3.720.1248

0.77.29.1

5.150.230.121212…

Irrational #s cannot be written as:

-Stop

-Repeat

-Have a pattern

-Be written as fraction

-Be a perfect square root

Examples:

√11√90√5

5.2187…0.1235…9.4321…