Simplifying/Expanding Irrational Radicals

What are radicals?

Just a fancy word for roots. (√)

What if the number under the radical isn't a perfect square?

It's going to be irrational, but as mathematicians, we must ALWAYS simplify. 

SIMPLIFYING IRRATIONAL RADICALS WITHOUT VARIABLES: 

√50 = ?

1. Find the greatest factor and a perfect square of 50!

The only perfect square that's a factor of 50 is 25.

 2. Take the square root of that exact factor!

√25 = 5

3. Put the "unchanged" number inside, and the "square rooted number" outside!

5√2 = √50 

EXPANDING RADICALS!

5√2 = ?

1. Square the number outside of the radical!

5² = 25

2. Multiply the result with the number inside the radical!   

25 x 2 = 50 

How can I tell if I've simplified enough?

Ask yourself, does the number inside the radical have any factors you can get the square root of?

If it doesn't (like having factors such as 7 or 5), then you have simplified enough! 

SIMPLIFYING RADICALS WITH VARIABLES:

For example,

√45x⁷ = ?

√45 * √x⁷ 

MAKE LIFE EASIER AND TREAT THEM SEPARATELY FOR NOW!

1. Simplify the constant.

45 = 9 x 5 

9 is a perfect square, now take the square root: 3

3√5 = √45

2. Simplify the variable.

√x⁷ = ?

UNLESS THE POWER IS EVEN, THERE SHOULD ALWAYS BE A NUMBER OUTSIDE! 

7 ==> 6  (Number NEEDS to be even, but we'll compensate for that -1 later, you'll see!)

Now take the square root of the "changed" power. 

√x⁶ = x³

THE SQUARE ROOTED THINGY IS OUTSIDE! 

The "LONELY" variable is INSIDE!

So...: x³√x 

3. Multiply!

3√5  * x³√x = 3x³√5x

Little note: "LONELY" means that the variable is to the power of 1, just meaning it's by itself. 

TRYING AGAIN, BUT WITH SOME EXPANDING!

For example:

2x⁴√6x³y²

1. That does not look right...

A VARIABLE THAT IS NOT "LONELY" AND IS INSIDE THE RADICAL...

Must ALWAYS be expanded, THEN, simplified.

2. Separate!

Treat them outside and inside as different groups!

(2x⁴)² * (6x³) * (y²)

3. Expand! 

Squaring 2 = 4

Squaring x⁴ = x⁸ 

4x⁸ * 6x³ * y² = 24x¹¹y²

3. Simplify! (You already know how this goes).

√24x¹¹y² = √24 * √x¹¹ * √y²

24 = 4 x 6 

√4 = 2

√24 = 2√6   

√x¹¹ = ?

x¹¹ ==> x¹⁰

√x¹⁰ = x⁵

√x¹¹ = x⁵√x

 √y² = y

 2√6 * x⁵√x * y = 2x⁵y√6x

Little note: The reason the y is OUTSIDE the radical is because we got it from a variable with an EVEN power, not an odd power. Also in this whole lesson, we are assuming the values are positive. They're not always necessarily greater than zero.

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