Simplifying Radicals + Pythagorean Theorem

Simplifying Radicals

Factor trees + Perfect Squares

What is a radical?

• Another name for a square root

Radical:

This is read as "radical 17"

Simplify "PERFECT" square  roots

A radical is considered a perfect square root IF when put in a calculator, it does not give you a decimal answer. (exceptions: see fraction slide)

Think of it  (sorta) as the opposite of squaring a number

Examples:

Using a calculator makes this easy

Answer is:

9

Some of them we know by heart.

Actually, it would be in your best interest to MEMORIZE the first 
15 perfect squares.
In other words, 1x1 = 1
2x2=4, 3x3=9,   4x4 = 16
 until you get to 15x15 =225

This way

You won't need a calculator to figure the square roots of those.
You will already know them.

If your radical is NOT a perfect square

Here is how you simplify them:

Factor Tree method

Factor

What numbers multiply to equal 50?
I chose 5x10

5 is prime but 10 is not. So I have to continue until all numbers (ends of the branches) are prime.

If there are pairs,

Circle them.
The square root of 5x5(25) is 5
That number goes outside the radical

Single numbers

Stay under the radical.
if there is more than one, multiply them together and put under radical.

Here is a more involved example

The prime numbers we have are 3,3,2,5.
The 3 will go to the outside of radical.  The leftover 2 & 5 will be multiplied and left under the radical.

Multiple Choice

√128

• 64√2
• 16
• 2√8
• 8√2

Multiple Choice

√200

• 14
• 20√8
• 5√3
• 10√2

Multiple Choice

• A
• B
• C
• D

Multiple Choice

• A
• B
• C
• D

Multiple Choice

• A
• B
• C
• D

Pythagorean Theorem

With radicals

Multiple Choice

The Pythagorean Theorem ONLY works on which triangle?

• obtuse

• scalene

• isosceles

• right

Multiple Choice

What side(s) will always be the longest on a right triangle?

• The Hypotenuse

• Leg A

• Leg B

• The Right Angle

Fill in the Blanks

Multiple Select

What side(s) will always be the shortest on a right triangle?

Select all that apply

• The Hypotenuse

• Leg A

• Leg B

• The Right Angle

Multiple Choice

What is the relationship between the sides in the Pythagorean Theorem shown?

Hint: Think about when the hypotenuse is missing!

• a + b = c

• a² + b² = c²

• a² – b² = c²

• a – b = c

Is it a Right Triangle?

Multiple Choice

Is this a right triangle?

• Yes

• No

Multiple Choice

Find the missing side.

• 18

• 16

• 17

• 19

Multiple Choice

Find the missing side.

• 11

• 12

• 13

• 14

Multiple Choice

Solve for x. SIMPLIFY your radical answer completely.

• 2√6
• 6√2
• 4√6
• 6√4

Multiple Choice

Solve for x. SIMPLIFY your radical answer completely.

• 4√2
• 2√4
• 2√2
• 16√2

Multiple Choice

Solve for x. SIMPLIFY your radical answer completely.

• √3
• 3
• √41
• 41

Multiple Choice

What is the measure of the missing side?

• √19 mi

• √23 mi

• 2 mi

• 3 mi

Poll

What is the best math/school meme?