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

Calculating Cost of a Car for Upfront Payment vs Monthly Payments

A situation that is probably useful for many people to understand.


$12000 cars for $8000 cash or $2000 down payment and $150 a month for 4 years. How much can you save by paying cash?


Find the total cost of the two options.

Option A: $8,000

Option B: $2,000 + ($150/month)(12 months/year)(4 years)

In option B, you add the initial cost to the rate multiplied by the time (with the correct units)

The find the difference between them.

How to calculate number of 16.9 oz bottles of water in a gallon?

Seems like you’re asking about how many of these water bottles it will take to have one gallon of water.

A gallon is a measure of volume, 231 cubic inches.

It would be much easier to calculate if you used the same kind of units for both the gallon and the bottles. So one option is have them either both in volume or both in weight.

I would personally figure out how many oz are in the gallon and then divide by the number of oz in each bottle.

You could do that in a few steps or if you understood the process, you could input a single query into Google and get an answer (which may need a little bit of interpretation).

Google is a little bit funny in what it accepts and does not accept.

If you input “oz in a gallon divided by 16.9 oz” which is exactly what you want, it doesn’t give you a direct result.

But if you input “oz in a gallon divided by 16.9” it gives you a number with units of oz.

For “oz in a gallon divided by 16.9 oz” the units will cancel out since you’re dividing oz by oz.

bottles in gallon

So the result is that number without the units of oz.

7.5739645 bottles.

Physics and Math Questions?

“How do I solve 2x-x=7?”

Combine “like” (same type) terms. You see three terms and and equals sign.

Two terms on the left side and one term on the right side (of the equals sign).

One term is a number and two terms have the variable x (with an implied exponent of 1).

If you had two numbers, you could combine them because they are “like” terms.

In this case, you have two like terms with x. So you can combine them.

Algebra, Multiplying by Reciprocal

The steps for solving an algebra problem, multiplying both sides of the equation by the reciprocal of the fraction on the left side.

multiplying by reciprocal

“why does $x^{1/2} = +\sqrt{x}$ not $±\sqrt{x}$?”

Seems you are asking about, in the context of differentiation, you’re not asking about how to actually differentiate.

I think you’re asking why it is not true that x^{1/2} = ±\sqrt{x}?

Since it seems logical that it would be similar to this, \sqrt{x^{2}} = ±x

Basically, I would say it’s because you can write -x^{1/2}

If that’s what you want to say, you put the negative sign in front of it. If you want it to be positive, you don’t write the negative sign.

If you square ±\sqrt{x}

In either case, you will get x.

“How do you Factor the following expression,[math] x^2-y^2-2x-2y[/math]?”

Here is how I would factor the algebra expression.

First thing I recognized was the difference between two squares. The more algebra you do, the more you will usually start to recognize patterns like this.

I used colors to show the connections between steps.

factoring with difference of squares 1

factoring with difference of squares 2

“How do I solve 4x+2y=-6 for y and then find two points on the graph” Algebra I

Solve for y
You want to solve for y =

So that means you should isolate y until it is by itself on one side of the = sign

You start with


Right now there are two terms on the left side of the =

4x and 2y

It would be better if you only had the term with y by itself

If you change the left side, you must make the same change to the right side

So your first step could be to subtract 4x from each side

Finding two points
Once you have an equation, you can pick any number you want for x and then determine what the y coordinate should be, by solving for y.

Each point has an x and y coordinate (x, y)

Or you could pick a y value and solve for x.

The values you pick should be easy to graph (probably near the origin).

Two values that can often be easy to work with are the y-intercept and the x-intercept.
You find the y-intercept by setting x = 0 and solving for y and you find the x-intercept by setting y = 0 and solving for x.