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a power, you raise both the numerator and the denominator to the power. X a-b =X X * O anything to the (-) power can be written over 1 a/b. 2 35=1 5⁰ = 1 -1562-1 the zero power anything to the zero power equals one. Negative exponents are the recipricals of the positive (integers) exponents. X ха xa - EXO KEO Xa XX-X0 - 1²8-170 Anytime you have a negative exponent, you can put 1 over that exponent, but positive. R 2-6=2° 2*6 = a X 0 Math Notes! 2-6 -5-(-1) use Keep change for exponents you can distribute the x-ponent outside of -5+1=(-4) the parenthysis through Hulti- placarion. when si exponents are different and bases are the same, you add the exponents. < Verse visa (54% - $2 (5².2³) ха = ( - ) ª have to 0.0001 7 0.001 4³ = 3 = 64 0·001 T Xa x-a 1 →→-→ → 8 1000 a fraction with an exponent as a deno- Minators equals to it's opposite Las a whole. T 4 -7° +7°. you you can not would do the problem because it does no exist for Solve/ there is no evaulate property the problem Math Notes numbers that are less than one, have a negative exponent a number is expressed in Scientific notations when it is written as a product of two quantities Ex. of a number in Scientific notation: <-3.106 <- 7.8-10 <-2.5-1070 -3-1073 -1.43.108 t It is no longer A Scientific notation after going past 10. power of 10 for 2 I and < 10 positive or negative exponents are oh in scientific notation. Smaller ex. Smaller ex. 180= 7.8·10 3025,000,000 = 3.025-10² larger ex. 2, 536,000,000 = 9.536 109 largest ex. 5,850,000,000 +5.85·10 equals 41, positive exponent equals 1 Smallest ex. 0.0000029 = 2.9·106 negative exponent pos exponent = go right neg" exponent = go left a specia way of writing very large or very small numbers. The number is greater than or equal to 1, but less than 10, and the second is a power of 10. 1. The distance from the earth to the Sun is approximately 93 Million Miles. write this number in sientific notation Step 1. Move the decimal point after the first non- zero digit, this will give you a number Between $10. Step 2. Count the number of digits you Moved the decimal point. This is the exponent of 10 in Scientific notation. Step 3. write the number in the form, A. 10 work: 93,000,000 = 9.3= 9.3.10 Math Notes Addition $ Subtraction, In Scientific Notation" (2.5+10¹) + (2.25 ·10°) = 25000000 + 225000000 = Step 1 Make the power of 10 the same Step a add /Subtract the co- efficients Step 3 put answer in Scientific Notation 250000,000 - 2.5-10% (2.5+10¹) + (2.25-10%) 2.5+10 +2.25-10¹ = Subtraction (5.97-1024) - (7.36·10 ²²) = 23 597-10-7.35-10 22 F Subtraction Addition is quiet Simple 597-7-35-10 D 589.65.10 = 5.8965+10 2.5+2.25-10¹ = 25.107 = 2.5-108 24 (8.2 -10%) (4.3 +0¹) - (8.2-4.3) (10¹ - 10¹)= = 35.26.10³ 3.5026-104 Express both numbers in Scientific Notation Remember: The product of powers property states that when Multiplying powers with the Same base, you add the exponents!" Addition: Use the Product of Powers Property Subtration: - Use the Product of Prowers Property. Division: - Use the Quitent of Powers Property Multiplacation: - Uses the Product of Powers Property I divide in scientific Notation 8.4.10³ 2.2 10-2 = (8.4 +2.2) (10²+10= 3.8181 105 add/Subtract in scientific Notation 6.1.104+ 218.10³ = 61+028-104= 6.38-10% Ex.1 (2.5·104)(1·10³) = 25·1·10³= 25-103 Ex.a 601H? 1.50% Math Notes: Meth addition lesson 1.37% check Journal Savaa's Journal 1.5 2.34% 2, 67100 3.70% 3.1/100. Per= 100 2. 12%. 3. 40% Convert -Convert Subtract in Scientific Notation (8.6·105) - (2.76·10³) = 860-2.76-103 = exponents Must be the same to add or subtract coefficents How Many times Greater Do you know how? 7. 99/100 an exponent goes left a decimal to go up and and tight a decimal to go down Scientific 857,84 -10³ = notation 8.5724-105 = Divison 8. a percent is a ratio that Compares a # to 100. % = per- cent 2.49% 3.30% 4.80% 9. By two 5.7/100 10. no 7 = 70%. bc 70= 70/100 Practice 1.31% 7,60% 2.28% 8.25% 3-551 9. 7%. 4.22% 10.50% 5.96% 6.12% H11 1st and when you have a slope, leave it as a fraction or whole number, turning it into a decimal is more difficult. proportional relationships go through the origin, Coo Step 1: graph (0,0) Step 2: Use slope to prot other pointis Math Notes positive slopes negative slopes rise = + rise ron = + y=3x Chart the points X 3x | y |(x, y) 3(0) 0 (0) 3(1¹) 3 (1,3) 4-3-3) rise run M=3 y = 3x use slope rise 3 1 -3-2 + 2 -4-3-2-1 3 - rise run 2 -4 either Method works 1 1st > 34 and X= rise run y=-2x * Jay 41(x, y) 021-10(0,0) 120-3(1₂-2) -12-12 1-1,2) rise run Y2-91 X2-X1 always write the slope in the form rise/run y = 1/2 x MQ = =/=/20 2 or -2 12-200M-2021 equations! One Saution. • No Solution - Infinite Solutions Ex. 3(2x+4)= 6(5x+2) J 6x+12=30x12 This is one Solution equation always remember to check if you have time Hath Notes -2442=121 Short Curs -24x500 M X = a number rather than X Ex. 8x* 3-10x = -2(x-2)+3 J No Solution -no value will Make the Equation true 3 = 4+3 L 347 1. X= 2 2.X* X 3X=X -2x +3 = -2x +4+3 V The ending final answer X = anything because the answer will Ex.3(x+1)+1+3x = a (2x+2)+X Still equal eachother will never be true. Infinite solutions Infinitely Many Solutions Infinite number of Solutions This is a no solution Problem 3 does not (=)7 Infinite Solutions -Infinite number of values will Make the equation true A -5 One Solution Only one Salation will Make the equation true ↓ 3x +3+1+2x = 4x +4+X ↓ 45x4 -5x ↓ (4=4 Ex. 2x=3 = x+1 Q If the variables are different, The equation will have ONE solution -8+3x=3x-8 if the Sides have the variables the same, there is an infinite number of Solutions -> This is an infinite Solution Proble H one 2x=x+4 Solution Problem 1x = 47₁ X=Y акт6 =5+2х If the variables are the same, then the problem either has no Solutions or infinite Solution because they Can be comple- tely removed -The point where a line crosses the y-axis of a graph The y intercept is also reffered to as the initial value when So, the initial value is the value of y x =O or the point where the line crosses the y-axis 3-2-1 essentially, it is the starting value of the relationship, it being the y = intercept example: example 2: 37 1 X Y O - 4 9 B 7 3 4 2 y-intercept intercept: 0 Slope :(0,0) 3 finding y-intercept Using a table -1 +1( 11 +1 the y-intercept is automatically -4 because it Shows = X when y is -4 X 1 3-1 2 0-1 example 2: S 2 3 ^ 4 Y 10 15 20 25 AY AX ya-yı ха-хі 2-5 )+5 {")+s ()+5 -=m y-intercept 5 Slope =5,1 -6-5-4-3-2 5-4 20 S 2 -1 - Jan 13, 2022 -4 -S -6V Slope = negative rise -2 =1=-2 run 1 I 12 3 4 S ya-yı Xa XI Slope : +2 yintercept: 4 m= Slope example 3: (-2,3) (-4,2) 9-3 -+(+2) 2 (-1/2) Slope is *also known positive as finding the change in rate whis equal to find- ing Slope. Math notes Midterm Topics Such as: Scientific notation Operations with Scientific notation Square roots Cube how to solve cube /Square root equations estimating non- n-perfect squares rational or irrational numbers. working with rational number operations. rules of exponents Find roots y- intercepts and Slopes From a line or graph Solving equations with variables on both sides Setting up and solving variable equations Combining like term graphing a propertional relationship from an equation Understanding Slope is the same as rate of change write a number in 0.00087 Properties ex.1 3²-3²=35 ex.a 8 6 Moth Midtery Review 1318.7.104 above one and below. zero > always of exponents! Raising a power to a power, Ex.1 Ex.2 6 24 (3³)-3¹² (4°) ³ = 4 12 Scintific notation -4 you Subtract & when Multiplying exponents with Ex.a the same base, add the exponents, when dividing, *when raising a power to a power, you Mutiply the exponents and keep the base Ex.1 (2.3-10 (30) (-2+4) (2.3.3.1) 2 (7.13-10³) Sanuary 18, 2022 (3.1-10-2) (2.3-104) (3.1+2.3) (-2-4) (5.3-8) J (5.3-10³)-(8-10²) (3-3) (4-5-10³) (1.35-1076) in oreder to add or Subtract your exponents need to be the same. Ex.3 Square Roots S16 = 4 √55=5 Jiw=1a Use Tone 54 56 54 113 COMManitive property or Standard formation ✓down right Teft * always check to Malhe sure. that they're in scientific notation Perfect Squares * Find Square root using perfect Squares 111 Additional Midterm Review 1 Square Bots Squares roots of a perfect Square are rational cational numbers -fractions -whole numbers -Terminating decimals *use ditributive property to Find graphing lines and for solving equations One Solution - x = anumber - (4=X) zera solution - no answer would make the equation correct Slope Many/infinite Solutions - Solution equals itself, 4=4 (X=X) Comparing rates of change rate of change = slope Slope = constant of proportionality y intercept - where y-axis is crossed when X=0 Jan 20, 20ƏƏ The graph of a line is G proportional if it is straight and goes, through (0,0) y = /hx = proportional h = slope use goes through (0,0) Finding Slope use rise run AY 4x = change in Y change in X linear equations in Srope intercepts form y=Mx + b slope ex.1 -S y MX (non- proportional relationship! (straight line = whether it contains B ar not y = MX + b | non portional 44 4. -1 Alegbra Study y intercept proportional graph Pr 5 3 any point along this line will Solve the equation _y=2x+1 * Pick at least 3 * right:1 up = 2 3 * write the equation in the Slope intercept form ex.1 3 -1 x 2x+1y|(x, y) 1 ex.a DECE ex.3 2(1)+13 (1₂3) 2(0)+1 2(-1)-1-1 (-1,-1) y=mx+b b=-2 M = 6 = y = 3x -8 M? b? 12 -8 Jan 25, 2022 y = 3x y=3x + 2 5 A when finding M. u are Finding Slope y=Mx J b=0 ALWAYS *when finding bu are finding y-intercept Straight lines * both are Straight lines; however one is proportion- al and one isn't no (x) = B as intial value X² in a formola is not a Straight line *The intersection of two unproportional not straight lines can help solve their equations S The Pythagorean theroem For a right triangle with legs a and b and hypotnuese C a+b=² = (a.a) + (b·b) = (C⋅c) longest Side • across the right angle Ex. 1 A 4 = √₁² -√20 X 6² +4²= 6² = a + 4 2 ex.3 Hath class 3 } >hypoten vese u a = √₂0 C= legth of the hypostnuese S. April 29,2022 3 1 5 2 3²³² +₂²² = √√C²² = √√₁3 = | C = 3.61 ex.a 3ª +4² = 6²³ 9+16=25 3,45 Theorem = 1² + 1²^² = c²ª 1+1=(² 3²+3²=4² 18 16 35 This not a right triangle x² = √6-√2 C = √₂ non-linear hures for horation Rotation 90° = positive angle of rotation ccw-c Ccw-counter clockwise (x,y)-> (-9,x) Rotation Rotation 180° (x,y)-> (x, y) Rotation 270° (² (x,y)-> (y,-x) cow-counter clockwise Sample. * Rotate 90° A(-4,3)-> A¹ (-3,-4) B(1,3)B(-3,-1) C(-1,¹)>('(-1,-1) Counter Clockwise April 1st, 2022 Ľ clockwise t your image will be the Same equadistance as the pre-image from your point of rotation # if asked to ratate again, point becomes a double prime #letter points don't change, they become primes A -> A -> x= non-linear double prime y=mx+b substitut. values when writing a linear equation! * Algebra intercept you will eventually be asked to write an equation For a line. real world problems 1. Rate = Slope -rate is always the X Value 2. Flat fee Fy- intersept - it never changes. Constant of proportionality is relatable to slope. ex-1 . . . ♥ January 31,2032 Solution: Si locate the y-intercept S locate another point that lies on the line 53cAlculate the slope from the y intercept to the Second point 54 write an equation in Slope intercept form given the slope and the y intercept you are visiting baltimore. A taxi Company charges a flat fee of 3 plus an additional $75 per Mile write an equation that you could use to find the cost of a rade. now Much for an 8 Mie ride MX+b y=.75, +3 Write a linear equation from two points Find the slope Substitute the coordinates Substitute the Slope for "M" graph an equation of the Form, y=MX Find the equation of the line Substitute Spolpe for "M" graph the line by plotting origin, then using slope plot" another point to form a line expanding, stretching, enlarging *reducing, Shrinking, downsizing *a dilation is a stretch or a Shrink of an image Dilations ex.a C ex.1 Da 0(-4,1) M (3, 2) G(252) *2₂ Da → 13 = D13 *Scale factor Usually a Fraction - enlargement * when stretching or shrinking, the image Stays proportional not a change in the proportion by which it changes orientation of shape M(-9,3) A (-3,9) *13 M²(-3,1) A(-1,3) C Dk April 5th, 20 -hos no change -reduction / Shrunk D = Dilation K= Scale Factor o'(-8,2) M(6,4) G(4,-4) streched to a larger Size when K is greater than 1, the image is (₂ Notation *when dilating an image with a scale factor, K? (x,y) →→→ (KX, KY) when I is equal to 1, the image stays the Same when K equals O, the figure is made to la Small to see when w is less than I and greater than 0, the image shrinks to a smaller Size Dilations result in an image that is larger or SHaller in size than the інаде not a change in size or orientation a transformation is a when translating a point: change in the position, Shape or size of a figure P(x,y). p'(x, y) P= Point P'= Point prime Image = 1 pre image after translation a translation is a transformation that Moves every point of a figure the same distance & the same P'(x,y) = P(x+a, yra) direction наде Translations Ex.1 ( P(3,0) Q(6-6 Translate a Line Segment 2 T-8,4 Points: LS 3 -3 ادا March 21, 2022 P(x,y)- pre-image" Points Prime: Plot the image of point P under a translation by 6 units to the left and 3 units up.. 2 • P 고 pre image P' (3-8, 0+4)= P'(-5,4) Q' (6+(-8)-6+4)= Q² (-2,-2) (x,y) V (3,-4) •P²(x+a, y+ a) image only Sliding one point (хтај уча) J R' 3-5₁-4+3) 2 (-2,-1) Pre image Ĵ units left 3 units down ех. 2. A: 1,2 B: S, I C: 3,4 is image A': -1, -1 6:3,2 c": 1,! TIL L write an equation from two points *find the equation a line that corresponds to both points ex. 2 (3,4) (5,8) 8-4 5-3 4 2 *£=x Math Classroom notes y = 2x +b y = 8=2(3)+b -2= b = Si calculate slope (S2) plug slope into the formula $3 use the x #Y -1=b of a given point in the formula to find " 11 (54) Salve for b $5 write the equation in (Y=MK +b) Form Try It Out!! > •_y=2x + (-2) 3=4+b x ex.3 C XI.YI X2-X2 *(4,5) (8,3) 5=0.5.4+5 5=215 ta 12 Jeburary 2nd, 2022 ↓ 3-5 -2=-0.5 8-4-4 Y=-0.5x+b 3=b y=0.5x+3 ex. 1 XIYI X2 Y2 (2,5)(4,13) 5 find Slope AY Y2-Y₁-8-4 = 4X Xa-Xi 2 13 5=4x +b 5=8+b -81-8 -3=b ex.4 * ход {1,2)(5,10) 10-a = S-1 4 80/5 Y=2x+5 10 = 2(5) +5 10=10+5 (0 =5) y = 2x ex.3 (30,20) 1. Solve one of the equations for either variable a. Substitute the expression from slep into the other equation 3. Solve the resulting equation 4. Substitute the solution from Step 3 into one of the original equations to find the other variable 5. write solution as an ordered pair x+y=50 2x+59=160 x + y =50 -y-y x= 50-y a(50-y) + 5y = 160 100-2y+Sy=160 100 +3y=160 -100 -100 3y =60 ÷3 Solve a system of equations by Substitution y=201 ex. 2 22 y=6x -11 y=1 (-2x-3y=-7 -2x-3(6x-11)=-7 -2x-18x+33=-7 12 (2,1) -20x +33= 33 -20x = -40 4-30 ÷-20 ex.5 y=-5x-17 -3x-3y=3 -3x-3(-5x-17)=3 --4 X=21 (4,3) -3x+15x+51 =3 ex.6 12x+51=3 -51-51 12x=-48 X=-4 March 14, 2022 ex.1 ex.4 y=x-1 ax-3y=-1 (2x-3(x-1)=-1 2x-3x+3 =-1 -IX+3=₂-1 -1x == 4 ÷-1 L X = 4 -¿² (²² 1 - 2x +-3x-3y=-15 y=3 =-14 X=1 elimination (4,3) y=-6 ex.7 & 4x+6=18 -6-6 -4x19=6 (-3₁-6) + -2 y = 2/ -4x=12 (3,-2) -9x = 27 -4 x=3₁ X=-3 LL Relate Solutions of Linear Systems y=x+4 y = -x +6 (1,5) = Solution) Solve- on a graph, were the lines intersect would be a solution for the equations in the system whatever point that is, will be the solve of both equations There can be -one solution -no Solutions - Many solutions > system of linear equations * a system of linear formed by 2 or More that use the same variables March 4,2022 Parallel lines will never intersect," therefore, there is no Salution Same Y-intercept Different Slopes = one Solution Same Slope + Many Same Slope C equations is linear equationst = Solutions no solution