Entire and Mixed Radicals
this week in math we learned about Entire and Mixed radicals and how to switch between the two along with reviewing our squares and cubes.
Entire radicals are composed of only a single number under the radical and no number beside the radical another way of saying it is the way that we have seen a number under a square root up until now. Entire Radicals tend to look more nice but aren’t as easy to calculate with. ex:
Mixed radicals tend to be easier to multiply or generally do math with. Mixed radicals look similar to how mixed fractions look like, with that they each have a base (the big number outside the fraction or radical) but the difference between a mixed radical and a mixed fraction is that with a mixed fraction the base (depending on the situation but for this example we shall use ) would be worth the total of the denominator, meaning that the 1 in if turned into an improper fraction would be , But for a mixed radical there is no base but in fact a coefficient which would multiply the root of said number. EX: would need to be turned into to be able to easily multiply the radicals together and once we multiply both radicals we should get an entire radical in the end and to turn an entire radical to a mixed radical you would need to figure out which square times another number gives me the number that you would have in this situation it’s 96. And then proceed to reduce the square into a coefficient to turn it into a Mixed radical.
Reviewing Real numbers and Learning If a radical is Irrational
This week in math one of the things we had reviewed what are the different categories of real numbers and we had learned how to identify whether a radical would be irrational or of the other groups.
Different categories of Real Numbers
The different categories of real numbers that we had reviewed were Natural Numbers, Whole Numbers, Integers, Rational Numbers and Irrational Numbers. Natural numbers are numbers that we would first learn, those numbers being anything above the number 1. Whole Numbers still include the numbers from Natural Numbers but this time only adding the number 0. Integers Includes both Natural and Whole numbers and adds to the table negative numbers (ex: …-3,-2,-1,0,1,2,3…). Rational Numbers includes all three of the previously mentioned number groups and adds a huge amount of numbers in between each of the previously mentioned numbers (ex: 0.1,0.2,0.3…) and could be put as a fraction. And now for Irrational Numbers. Despite being part of the Real Numbers Irrational Numbers are not part of the last 4 groups (Natural, Whole, Integers, Rational). Irrational Numbers are numbers that cannot be divided easily or are just plain hard to calculate with. For a number to be irrational the number must have the following criteria: must be a decimal that does not terminate or have repeating numbers. A few examples of Irrational numbers would be , (PI). These two are just a few of the many irrational numbers that are out there. It’s now time to apply this information to know if a radical is irrational.
To know if a radical is irrational we must first know which numbers are squares or cubes. 4*4=16, 4*4*4=64, 3*3=9, 3*3*3=27, 2*2=4, 2*2*2=8. To keep it simple let’s say the radical is upon looking at our list we would be able to see that this radical would be rational as it is 2 because we already know that 2*2*2=8 making it rational. but if the radical was it would be irrational as it is not on our list of squares and cubes.
Arithmetic Sequences: Arithmetic Series
This week in math we expanded on Arithmetic Sequences having learned a new equation.
The equation in question that we learned this week has to do with a series. And to explain what a series is in this context: a series is a specific amount of numbers in a sequence, so let’s take the example of 1,2,3,4,5…100 this sequence has 100 terms in it and thus it is a series of 100, but you might be wondering “what does this have to do with the equation in question?” and to that I respond with it has everything to do with the equation. Now that we have an understanding of what a series is we can move on to the equation. the equation is: Sn=n/2(t1+tn), in this equation the n is replaced by the last term, in this case, is 100 making the equation look like this S100=100/2(1+100) –> S100=50(101) –> S100= 5050
Arithemetic Sequences: Equations for finding the First Term, General Rule and Difference.
This week in math we learned a few equations for finding out the first term, general rule and difference to help find out if a specific term is in the sequence.
The equation that we learned is: tn=t1+(n-1)(d) in this equation the tn is the number that we are looking for or the final term in the sequence. The n stands for where in the sequence the specified term is whether it be 25 or 4. The d in the equation means the difference between the terms if it is given to us, so in the sequence: 1,3,5,7… the difference would be 2 since the first term increases from 1 to 3 and the second term increases from 3 to 5. Given that same sequence if we try to find the 25th term in the sequence it would take quite a while if we were to just add 2 to the sequence each time until we find the 25th term in the sequence, so we use the equation to help us find it much faster. So the equation would look something like this if we are to use the example from before and after inputting each of the hints in their respective places: t25=1+(25-1)(2) then we simply do some distribution so there are no brackets.(t25=1+50-2) then we add all the numbers together so that t25=49.
Solving systems using different methods
This week in math we learned about a few different ways of solving a system to find the solution.
Solving using Inspection
The way that you solve with inspection is by literally guessing numbers that make sense for both equations. This method is the hardest to use and doesn’t help with solving at all due to there being many different answers and it would just take ages to solve.
Solving Using Substitution
Solving using Substitution is my favorite way of solving for a solution as all you do is isolate either the x or y of one of the equations (it makes it easier when the x or y doesn’t have a coefficient example:[x+y=-5] in this example all you need to do is move either the x or the y to the other side of the equation) after this step you take the equation ex:x=-y-5 and input it into the other equation by replacing the x in the other equation for the -y-5 example of the second equation being: 3x-5y=9. By inputting the first equation into the other one it would now look like this: 3(-y-5)-5y=9 then you just let algebra take over and solve for y. But we are not done yet, after solving for y we need to input that into the equation we got the y from to solve for x. With y=-3 we look back into the first equation and input that to where the y is x=-(-3)-5 then we do algebra again to get x=-2.
This week in math we learned about what a system is.
So a system has a few parts to it. It consists of having always TWO LINEAR EQUATIONS, and zeroone, two or infinite solutions. And to explain what a solution is, a solution is the point where if put on a graph the two lines touch(Intersecting Point). If they have zero solutions they are parallel lines and will never touch at all or you could say that they have the same slope but not the same y-intercept. If they have one solution both of the lines have a different slope and may or may not share a y-intercept. We have yet to learn about when they have two solutions. Finally when they have infinite solutions it just means that they are the exact same line touching every where, meaning they have the same slope and y-intercept.
The different equations of a slope
This week in math we learned three different equations for a slope and how to change them into other forms if possible.
the equation that we learned for this is y=mx+b the m in the equation is the slope or the rise and the run. The b in the equation is where on the y-axis that the slope intercepts or you could just say that the b is the y-intercept. An example of this equation would be y=3/10x+5. with the 3/10 being the slope and the 5 being the y-intercept.
With point-slope form the equation is m(x-x1)=y-y1. The x1 and y1 are a coordinate that has been given to us so that we can solve the equation. And again the m is the slope. This equation is used to be shown as an equation and it’s really easy to turn it into Slope Y-Intercept form just by doing a bit of algebra.
General Form (AKA Pretty Useless form)
The equation for general form is always ax+by+c=0. One of the things that needs to be true is that the x cannot be dividing by anything. This equation is also known as being pretty useless because we cannot tell a single thing about the slope when it is in this form, yet this equation makes it really simple to input it into a computer program and doesn’t look too complex. The only way to get informztion about the slope through this form is by doing algebra. And one of the ways to figure out whether or not a specific point is on the graph is by inputting them in their respective places by the x or the y. A way to check the x and Y-Intercepts is to either remove the x or the y depending on which one you chose to solve for, then moving the constant term to the other side of the equation and dividing so that the x or y is on it’s own.
The slope of a Line
This week in math we learned how to make find the slope of a line.
To find the slope of a line we learned the equations rise/run and y1-y2/x1-x2 of which the first is used when you already have the slope and are trying to find where the next nice point (a point on a line that is not in between any of the squares on a graph) on a line is. A really easy way that we learned to remember the equation is you have to rise up before you can run. the rise part of the equation means how many units that you go up or down on the y-axis and the run part of the equation tells you how many units you go right or left on the x-axis. You could say that it’s like a set of directions. Take the example of 3/2 with our starting point being (0,0). the next nice point on the graph would be (2,3) then (4,6) and so on so forth. The way we got to there was by going up 3 and going over 2. But what if we didn’t have a slope but instead two points on the graph (3,1) and (-15,25) in this circumstance we need to find the slope, and the equation that we learned to use is y1-y2/x1-x2. so what we do first is to take both of the y-coordinates and subtract them as follows 1-25=(-24), and do the same with the x-coordinates: 3-(-15)=+18. so now the equation should be -24/18 and the only thing we do from here is to simplify down to -4/3 by dividing both sides by 6. The slope -4/3 is a negative slope (a slope that goes down instead of up)