Author: Bon Crowder

How and When You Can Manipulate Square Roots

Square Roots (aka radicals) are part of the multiplication world. They have the same properties, but because they look weird – living in little “houses” – it is hard to grasp what they can do.

This is how the “houses” (the radicals) can “move”:

After that, you can work with multiple radicals:

It is important to note that this little trick works because it’s all multiplication, division and roots. If you have addition or subtraction in there, you’ve got yourself a challenge. Watch this to see how to deal with addition in the mix:

Thanks to jrobbo at MyMathForum.com for the inspiration for this post.

February 9, 2011
How to Add and Multiply in Base 12

In the previous post, I explained some basics about base 12 counting. Since you’re back, clearly you’re a glutton for punishment (or your Algebra book is requiring you to do this).

Either way, here’s how to add in base 12:

And how to multiply:

I cheated a little here here, because to me this is the easiest way. If you need an example or two to do it “right,” post it in the comments and I’ll do a few for you.

A big thanks goes out to Kellie for the question.

February 7, 2011
What Base 12 Means

Algebra books seem to believe that base 12 is a good idea for humans to use. This is only in the case of that pianist in GATTACA that has 12 fingers. We have a base 10 system because we have 10 fingers. (This is why you’re allowed to count and do arithmetic on your hands.)

Regardless of the general uselessness of a base 12 system, working with it is a good exercise in patience, alternate thinking and puzzle solving.

Here’s how to “count” in base 12:

But what if you want to get higher than you can count? For instance, what if some algebra book asks, “What’s after EE0₁₂?”

Now check out how to switch a base 10 number (which is our way) into a base 12 number:

Want more? Check out how to add and multiply in the next post.

Thanks to Kellie for the inspiration for this post.

February 7, 2011
What the Heck Is the Zero Product Rule?

The fancy answer is, “For all numbers a and b, ab = 0 implies a = 0 or b = 0.”

The less fancy, and still strange answer is: If two things multiply to get zero, one or both of them better be zero. Like this:

Too much pixie dust, for my taste. What doesn’t work, for sure, is this:

So why does it work with zero and not three?!? To answer this, we need a little background on fractions and zero. And pizza – pizza’s always a good thing to add to math:

Using the fact in the video above, the “foorp” in this video shows why it’s true:

Special thanks to GFC at MyMathForum.com for inspiring this post.

February 4, 2011
Those Pesky Exponents – How to Distribute Exponents over Multiplication

Remember that you can distribute multiplication over addition? You can also distribute exponents over multiplication.

Here’s how and why it works:

And here it is written out “long ways” for an algebra problem. Remember, you can use the short version. I do the videos with the long version to help you remember why it works.

Have any questions? Post them in the comments!

February 2, 2011
2 Ways to Multiply Mixed Numbers

Multiplying fractions provides an interesting challenge. Put some mixed number into the situation and you could be facing a real bear.

Here’s one way to work the problem $2 \frac{3}{4} \times 8 \frac{1}{7}$ by converting it to mixed numbers:

Or you can use the handing dandy distributive property (or as some might recognize it – FOIL):

Here’s one more for fun:

Thanks to NapNinja4 on MyMathForum for this question.

February 1, 2011
How to Convert Mixed Numbers to “Improper” Fractions

This isn’t something you do everyday. Therefore it would be helpful to know how to remember it long term.

The mixed number $2 \frac{3}{4}$ really means $2 + \frac{3}{4}$ .

Notice the PLUS sign? That’s the big key to doing this.

This video shows the long expanded way that most people won’t show you. You wouldn’t really convert this way, but it helps to see it in the long form. This way you know what your teacher is showing you isn’t some pixie dust magic:

Now that you can do that, you can probably go the other way too:

And how about checking your work? Nevermind the devil box (aka the calculator), the more you check your own work by hand, the more your confidence in your own math abilities will be. Try doing this:

What do you think? Will this help you remember? Do you have another way to remember? Share it in the comments.

January 31, 2011
How to Use the Quadratic Formula to Never Factor Polynomials Again

We learn to factor polynomials so we can solve for x. Stuff like

3x² + 2x – 1

can be easily factored into

(3x – 1)(x + 1)

But some things aren’t so easy. And some things are just down right a pain in the bottom. Like this one:

18x² + 189x + 490

“Just shoot me,” you might be thinking. But cool your jets, cowboy. Here’s the first tool you need to never factor polynomials again:

I know – who wants to memorize that formula? But wait. If you use that one formula that you can memorize, you never have to factor polynomials again. Watch:

So the factoring isn’t bad on that one, right? How about this one:

And how about if you come across one of these. Wouldn’t it be nice to get rid of all guesswork:

This has some disadvantages, of course. You see that there’s a lot more arithmetic. And there’s some things to look out for. But if you hate trial and error (like I do), then you might be willing to take the bad with that good.

Whatcha think?

January 28, 2011
How to Multiply By Hand Without Hurting Your Brain

There’s an amazingly fancy tool that I began discussing in the last post. This D-word probably has been a close friend of yours for a while. It’s likely you’ve used it before and just didn’t realize it.

Well, now you can give it a name and give it some practice. It’s the Distributive Property.

Fancy name. Fabulous friend.

Check out what’s really going on with 87 x 48 here…

That one you can do quickly with pencil and paper. Here’s an easier one that you are more likely to do in your head (after practice with P&P of course)…

Have you done this before? Share your stories in the comments.

January 26, 2011
How and When to Cancel Things in Fractions

One of the common errors in algebra is canceling bits in fractions that can’t be canceled. To remember the rule, though, you don’t have to memorize anything. You only have to think of a plain arithmetic problem to compare it.

To help you remember what’s allowed with problem:

$\frac{3x+4}{2}$

think about this simpler problem:

$\frac{8+4}{2}$

Like this:

And just to keep you straight, here are the requirements (and a couple of examples) for the fancy <cue music> “distributive property.”

What do you think of this explanation? Post your thoughts in the comments.

January 25, 2011