“What is the meaning of “x over y” in mathematics? Does it mean x /y or y/x?”

x_over_y

x is above y in the fraction, hence “x over y”. If you use a / then the two variables are side by side. I almost always recommend using the numerator above the denominator way of writing fractions instead of the slash mark, generally much more functional.

Getting Points from a Linear Equation

Let me know if you have questions.

Square root of 125 simplified

Square root of 125 simplified

First I factor what is within the square root. You could say it’s √(25*5) and then factor again, I went instead to √(5*5*5)

From there you can separate the two pieces in their own square root signs.

Then you have a whole number multiplied by the square root of a prime number.

Simple Factoring, Quadratic Polynomial, Algebra


Factoring a simple quadratic polynomial.

(assuming you can factor it)

First start with two sets of parentheses. Then I look at the first term then the third term and finally the second term.

Synthetic Division of Polynomials

We started by looking  at long division and synthetic division of polynomials.

Long division was familiar, synthetic division was new.

Synthetic division can be faster and less written work, but is more straightforward with certain conditions. You can use one method to check the other.

There is a decent explanation here, but with a mistake in the final example.

http://www.mesacc.edu/~scotz47781/mat120/notes/divide_poly/synthetic/synthetic_division.html

The ‘test zero’ goes on the outside. If you are dividing by (x-2), the test zero is 2, if you are dividing by x+6, the test zero is -6, etc. The test zero is the number that makes the denominator zero when you plug it in.

Synthetic division is straightforward when the coefficient for the variable in the denominator is of the form x + a and the leading coefficient of the numerator’s first term is 1.

Wikipedia describes how you can use it when things get more complicated, but I would probably just use long division if that is the case.

We then looked at some graphing of equations and inequalities. To find the x-coordinate of the vertex of a parabola you use -b/2a.

Also simplified some expressions.

Fractional Exponents, Logarithms, Quartic and Cubic Equations

We started by looking at some current homework which involved factoring. For most of the problems they started with a quartic or cubic equation and then factored into a quadratic and one or two other terms.

Cubic equations can have three solutions, quartic equations can have four.

Talked a little about one application of imaginary numbers since we saw complex numbers in some solutions. And went into more detail about how to do calculations with them.

Talked about fractional exponents for a bit and how to work with them. Also about logarithms. The base being 10 if no other base is shown.

Fractional exponents are almost always easier to work with than ‘cube roots’, ‘fourth roots’ etc. Square roots aren’t too bad but still can be harder to manipulate in some cases.

Last problem we worked on was a three dimensional volume of a cone with the top cut off. Looked at the application of similar triangles and the proportionality of the sides.

“What is the square root of 96 in radical form?”

Like Terry Moore said, it actually is pretty easy to work with in the original form,

√96.

Sometimes it is better to ‘simplify’ by getting a coefficient multiplied by a smaller number within the radical. Other times, you’re really not simplifying, but making it more complicated. This is probably one of those times where leaving it as √96 is better.

If you had something like √196, you would want to change that form since it turns out that 196 is a perfect square.

Regardless, I’m assuming this is a problem for a class and your teacher/text book wants you to do the following……

You could start by factoring it in small steps within the radical sign.




If you find factors that are perfect squares, you can take them out of the radical.

You can find large factors if you want, but you can also start with small factors.

Probably the easiest number to factor out is 2 since 96 is an even number.

96/2 = 48.

square root 96 radical form

48 is also an even number, so you can factor another 2.

Now you have √[(2*2)*24]

If you can factor out larger numbers, it can save you a little time.

So from here let’s factor out a 4.

√[(2*2)*(4*6)]

= 4 √6

What is the square root of 5,000,000?

Saw this question which was asked at a job interview for an engineer on Quora,

My response:

It’s often useful to be able to make approximations quickly.

If you make a somewhat accurate fast approximation, sometimes you can tell whether something is a good idea or bad idea without investing much more time.

With an engineering background, it might be assumed that you have done many calculations and are pretty good at making them.

Many people have done a lot of calculations without a calculator, Google, Mathematica, etc.

They probably wanted to see your response to the question as well as getting an idea of your sense of numbers.

If you said 42, in this case, it would not be the answer to get you a job. Though it might get a laugh.

Saying 1,000 would be somewhat reasonable and the right order of magnitude since 1,000^2 is 1,000,000.

10,000 would be too big and not the right order of magnitude.

2,000^2 would be 4,000,000 so it’s getting more accurate.

If you knew that 1,000^2 is 1,000,000 and that you would still need to get the root of 5 and said that the answer is exactly √5 * 1000, the interviewer would probably be impressed.

If you said something that was an approximation of that and pretty close, that would probably also go over well.

 

“How to find the square root of 196?” (Square root of 196 simplified)

You could definitely use a calculator or Google.

But if you would like to do this by hand, you can without too much trouble.

Written explanation continues below the video.

It’s possible you recognize that 196 is a perfect square. If you do, but you’re not sure what it is, 196 is definitely an even number. So you can divide by two.

It turns out that 196/2 is 98, which you can also divide by two.

So 2 x 2 x 49 = 196. You should recognize 49 as being a perfect square.

square root of 196

So the square root of 196 simplified is 14.

And it’s good to know that you can square a negative number and the square will be positive.

The convention of saying “the” square root means one, but it’s good to know about multiple solutions.

 

Additional Problems:

And some numbers that aren’t perfect squares,

Try finding

1. The square root of 20

2. The square root of 228

3. The square root of 52

“What is the square root of 28 over square root of 7?”

square root 28 over square root 7

Here’s one way to do it, factor the square root of 28, completely and then go on from there.

Another method is a bit less work,

You could place both under the same radical sign, simplify, then solve

square root 28 over square root 7 faster