Algebra (Period 5)

Simplifying Radical Expressions 10-2

SQUARE ROOTS (RADICALS)

MAIN CONCEPT:

Square rooting "undoes" squaring!   It's the inverse operation!!!... Just as subtraction undoes addition     Just as division undoes multiplication

If you square a square root:
(√243)² = 243 (what you started with)

If you square root something squared:√243² = 243 (what you started with)
If you multiply a square root by the same square root:
(√243)(√243) = 243 (what you started with)

IN SUMMARY:

(√243)² = √243² = (√243)(√243) = 243

1) RADICAL sign: The root sign, which looks like a check mark.

If there is no little number on the radical, you assume it's the square root
But many times there will be a number there and then you are finding the root that the number says.

For example, if there is a 3 in the "check mark," you are finding the cubed root.
One more example: The square root of 64 is 8. The cubed root of 64 is 4. The 6th root of 64 is 2.

2)RADICAND : Whatever is under the RADICAL sign
In the example above, 64 was the radicand in every case.

3) ROOT (the answer): the number/variable that was squared (cubed, raised to a power  to get the RADICAND (whatever is under the radical sign) In the example above, the roots were 8, 4, and 2.

4) SQUARE ROOTS: (What we primarily cover in Algebra I) The number that is squared to get to the radicand.

Every POSITIVE number has 2 square roots - one positive and one negative.
Example: The square root of 25 means what number squared = 25
Answer: Either positive 5 squared OR negative 5 squared = 25

5) PRINCIPAL SQUARE ROOT: The positive square root.

Generally, we assume the answer is the principal square root unless there is a negative sign in front of the radical sign or you’re solving to find both roots for a quadratic (We’ll deal with that in Chapter 13-2).

6) ± sign in front of the root denotes both the positive and negative roots at one time!

Example: √ 25 = ±5

7) ORDER OF OPERATIONS with RADICALS: Radicals function like parentheses when there is an operation under the radical. In other words, if there is addition under the radical, you must do that first (like you would do parentheses first) before finding the root.

EXAMPLE: √ (36 + 64) = 10 not 14!!!!

First add 36 + 64 = 100

Then find √100 = 10

Radicals by themselves function as exponents in order of operations
(that makes sense because they undo exponents).
Actually, roots are FRACTIONAL EXPONENTS!
Square roots = 1/2 power,
Cubed roots = 1/3 power,
Fourth roots = 1/4 power, etc.

So √25 = 25^½ = 5

EXAMPLE: 3 + 4√25
you would do powers first...in this case square root of 25 first!|
3 + 4(5)
Now do the multiplication
3 + 20
Now do the addition
23

8) THE SQUARE ROOT OF ANYTHING SQUARED IS ITSELF!!!

EXAMPLE: √ 5² = 5

√ (a -7)² = a - 7

RATIONAL SQUARE ROOTS: Square roots of perfect squares are RATIONAL

REVIEW OF NUMBER SYSTEMS:

Rational numbers are decimals that either terminate or repeat

which means they can be restated into a RATIO a/b of two integers a and b where b is not zero.

Natural numbers: 1, 2, 3, ... are RATIOnal because you can put them over 1

Whole numbers: 0, 1, 2, 3,....are RATIOnal because you can put them over 1

Integers: ....-3, -2, -1, 0, 1, 2, 3,....are RATIOnal because you can put them over 1

Rational numbers = natural, whole, integers PLUS all the bits and pieces in between that can be expressed as repeating or terminating decimals: 2/3, .6, -3.2, -10.7 bar, etc.

Real numbers: all of these!

In Algebra II you will find out that there are Imaginary Numbers!
Square roots of NEGATIVE numbers are IMAGINARY

IRRATIONAL SQUARE ROOTS:  Square roots of a nonperfect squares are IRRATIONAL -

They cannot be stated as the ratio of two integers -
As decimals, they never terminate and never repeat -
you round them and use approximately sign.

MOST FAMOUS OF ALL IRRATIONAL NUMBERS IS PI!

  

note-- Square roots are MOSTLY IRRATIONAL!

There are fewer perfect squares than nonperfect!

Here are some perfect squares: 0, 4, 9, 16, 25, 36, etc.

PERFECT SQUARES CAN ALSO BE TERMINATING DECIMALS!

EXAMPLE: √( .04) is rational because it is ± .2

But all the square roots in between these perfect squares are IRRATIONAL

For example, the square root of 2, the square root of 3, the square root of 5, etc.
You can estimate irrational square roots.

For example, the square root of 50 is close to 7 because the square root of 49 is 7.
You can estimate that the square root of 50 is 7.1 and then square 7.1 to see what you get. 

If that's too much, try 7.05 and square that.
This works much better with a calculator! 

And obviously, a calculator will give you irrational square roots to whatever place your calculator goes to.
Remember: These will never end or repeat

(even though your calculator only shows a certain number of places physically!)

SIMPLIFYING RADICALS

SIMPLIFYING NONPERFECT NUMBERS UNDER THE RADICAL:

A simplified radical expression is one where there is no perfect square left under the radical sign
You can factor the expression under the radical to find any perfect squares in the number:

EXAMPLE: √50 = √(25 * 2)

Next, simplify the SQRT of the perfect square and leave the nonperfect factor under the radical:

√(25 * 2) = √25 * √2 = 5√2

We usually read the answer as "5 rad 2"

HELPFUL HINTS:
When you are factoring the radicand,
you're looking for the LARGEST PERFECT SQUARE
that is a FACTOR of the radicand.
So start with:
Does 4 go into it?
Does 9 go into it?
Does 16 go into it?
Does 25 go into it?
etc.

Another method: Inverted Division or Factor Trees
Factor the radicand completely into its prime factors (remember this from Pre-Algebra?)
Find the prime factorization either way in order from least to greatest.
Circle factors in PAIRS
Every time you have a pair, you have a factor that is squared!
Then, you can take that factor out of the radical sign.
Remember that you are just taking one of those factors out!

Example: √ 250
Prime factorization = 2 x 5 x 5 x 5
Circle the first two 5's
5 x 5 is 25 and so you can take the square root of 25 = 5 out of the radicand
Everything else is not in a pair (squared) so it must remain under the radical

Final answer: √ 250 = 5√(2 x 5) = 5 √10

If there is a number in front of the radicand, simply multiply it by whatever you take out.