7th grade

What is interest?

Interest is a percentage rate that is paid on borrowed money.

Sometimes, that interest is paid by a person to a lender or bank (when that person borrows money).

Other times, that interest is paid to a person by a lender or bank (when that person deposits money that the bank can then lend to other people).

When someone borrows or deposits money, they are given an interest rate (usually in percentage form).

The amount that they borrow or deposit is called the principal.

What is a commission?  People earn a commission when they earn a percentage of what they sell.  Many professionals (e.g., realtors, salespeople, mortgage brokers, etc.) work on commission. Every time they make a sale, they get some percent of the sale prices as their salary. That percentage that they earn is a commission.

How do you calculate a commission?  

Someone might say that his commission is 4% of the sales price.  

Although it's not at all hard to increase or decrease a number by a percent (review the Percent Increase/Decrease lesson if this doesn't sound familiar), it can be tricky to find the percent a number was increase or decreased by.

Finding a percent of a number means finding a part or portion of a number. Increasing/decreasing a number by percent means finding a percent of a number and adding or subtracting that amount from the original number. We deal with percent increases (mark-ups, tips, sales tax) and decreases (sales, discounts) all the time. Calculating percent increases and decreases mostly involves finding a percent (if you need to review that, review the Finding a Percent lesson) and then adding it to, or subtracting it from, your original number. 

When you multiply a square root by itself, the square root disappears.

$\sqrt{7} \times \sqrt{7}=7$

This leads to a pattern when working with $i$ (which equals $\sqrt{-1}$):

$i^1=\sqrt{-1}$

$i^2=\sqrt{-1} \times \sqrt{-1} =-1$

$i^3=\sqrt{-1} \times \sqrt{-1} \times \sqrt{-1} =-\sqrt{-1}  $

$i^4=\sqrt{-1} \times \sqrt{-1} \times \sqrt{-1} \times \sqrt{-1}= -1 \times -1 = 1$

In math, you can divide just about everything, including radicals!

But, let's remember, we don't put fractions or decimals under radicals (in final answers), so we only divide radicals when the number that will end up under the radical is a whole number.

When radicals are divided by other radicals, you can divide the numbers and keep the product under the radical. If you have a hard time remembering this rule, you can always test the rule with square roots that you know. 

In math, you can multiply just about everything, including radicals!

Let's review what a radical is: a radical is a root.  If there's no superscript number in front of the radical, it's a square root (the 2 in front of the radical is assumed if there is no other number), which asks "what number, times itself, equals this number?"

Because a radical is a square root, a radical squared is the number under the radical:

$\sqrt{12}^2=(\sqrt{12})(\sqrt{12})=12$

You already know that when a number is under a radical (e.g., $\sqrt{49}$), you're supposed to find the square root of the number (the number that, when multiplied by itself, equals the number under the radical).

You also know that most numbers are not perfect squares, so it's nearly impossible for the human mind to calcluate most square roots.

However, we can simplify many square roots -- even if we can't calculate them precisely.

Scientists often have to deal with very big numbers (how far is it from Earth to the Andromeda Galaxy?) or very tiny numbers (how big is an atom?).

As we know, big numbers have lots of zeros at the end (it's 2,538,000 light years from Earth to the Andromeda Galaxy), and small numbers require lots of zeros after the decimal point (some scientists estimate that atoms are about .00000001 centimeters in diameter.)