Pre-Algebra

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.

Finding the square root of a perfect square is easy.  What number, times itself, equals your number?

$$\eqalign{\sqrt{81}&=9\\\sqrt{400}&=20}$$

But, what do you do when you need to find the square root of a non-perfect square.

If you have a calculator, it's simple, you use the $\sqrt{\text{  }}$ button.

If you don't have a calculator, you need to estimate.  You won't get a perfect answer by estimating but you can come close. 

It's helpful to think of square roots as the opposites of squares.

The square of a number is what you get when you multiply a number by itself.  The square of $6 = 6^2 = 36$

The square root of a number is the number that you have to multiply by itself to get another number.  So, the square root of $36 = \sqrt{36} = 6$

$6 \times 6 = 36$ so the square root of $36 = 6$.

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.)

When dealing with very big or very small numbers, it's often convenient to compress them and make them uniform.  We use scientific notation to do this!

When writing in scientific notation, you move the decimal of any number to just after the first (highest) non-zero digit.  You then multiply that number by a multiple of 10 to expand it back to standard form. For more on the details of how scientific notation works, see Scientific Notation (Write in Standard Form).

What does that look like?

Ratios can look complicated, but they are just fractions that show the relationship between two numbers that increase and decrease together.  You can write ratios with colons (6:5) or as fractions 6/5. 

Math has a defined and precise order of operations that must be followed for problems to come out correctly. Calculators are programmed with the order of operations. Students have to learn them!

Most students learn the acronym PEMDAS to remember the order of operations.  PEMDAS stands for:

1st do Parentheses (do operations in parentheses first -- following PEMDAS inside the parentheses!  Do nested parentheses from the most nested (deepest inside the parentheses) to the least nested)