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√(189)           

3

First, we must see if there's any 

perfect cubes it"s divisible by. You 

can make a list by typing "x^3" in 

your calculator in the y= section 

then do "2nd graph" to pull up a 

list like this:

Now then, let's try 27.

√(27)* √(7)

3

3

189/27 = 7

it works

√(27)* √(7)

3 √7

3

3

 3

The perfect cube of 27 is 3 and 

since   √(7) has no perfect cube, it 

remains in the radicand.

Therefore it simplifies as:        3 

√7.

3

It's actually a very simple rule we must follow. We can only 

divide/multiply things in the same place. For example.

2√   *   3√

6√      is the 

answer we get. 

Now, we must 

reduce this.

12

In this situation, we can only multiply 

by what's inside the radicand and what's 

outside: 

Outside: 2*3 = 6

Inside: 3*12 = 36

36

36/4 = 9

6 * √    * √    = 36 final answer

4

9

The rule that we must only div./mult. 

numbers in the same place applies to 

division as well 

Only mult./div.:

Inside * Inside/Outside * Outside

3

Adding and subtracting 

radicals is a like regular 

addition and 

subtraction.                                                                                                                                                                                           

The first thing to 

do is to make sure 

that what you are 

adding or 

subtracting have 

like terms.

Make sure they 

have the same 

radical. Next, you 

add or subtract the 

coefficients, not 

the radicals.

Finally reduce your 

answer and you're 

done! Here's an 

example:

Adding

1) √7 +4√7

                  = 

1√7+4√7

                  = 5√7

2)√36+√64

                  = 6+8

                  =14

The first problem has 

like terms, making it 

easier to add. The 

second problem has 

two perfect squares, 36 

and 64.

Subtracting

1) 6√8-2√8-√44

              =4√8-2√11

           =-2√11+8√2

2) 2√20 -4√5-4√20

                    =-4√5 -4√5

                    =-8√5

Like the addition 

examples, these problems 

also have like terms, 

making subtracting easier 

as well.

Multiplying Binomials

When you multiply binomials with a radical, it 

is the same as if you were multiplying without 

it.

Example: (-3-5√3)(-1+√3)

-3-5√3

-1+√3 *

_________

-3√3-5√3

-3√3-5√9

_________

-12-8√3

Rational exponents

Rational exponents are another way to right a radical 

expression but just look like a fraction in the 

exponent's place

_

25

1

2

In this example, I will show you how to read 

this problem and how to put it into radical 

form.

Now some basic rules for these types of 

problems are simple.

Numerator=the number's power

Denominator=index

integer= what's in the Radicand

so with these rules you would write it like this:           

Now this is technically correct, but math 

notation dictates we don't need the index of 

2 or the power of 1 to show. so your real 

answer is 

25

√

2

1

√

25

Now the other way around:

√

5

24

2

So all we need to find is the index and see if the 

radicand goes any certain power as well as what the 

radicand is.

index=5

power=2

radicand=24

When we put this all together we get:

2

24

_

5

and this is the correct answer.

Another way to write this would be 24   

Don't be caught off guard when you come 

across this as it's just a fraction in its 

decimal form.

.4

Keep in mind these rational exponents are 

still well.. EXPONENTS so the same rules/ 

laws apply to them

ex)

9        9      =    9     + 9    =   9    = √

_

_

1

1

2

4

.

_

_

1

1

2

4

4

9

4

3

When you multiply exponents with like bases 

you must add the exponents and if you divide 

them you must subtract the exponents. 

Bottom line ALL LAWS OF EXPONENTS APPLY 

TO RATIONAL EXPONENTS!

_

3

1.Product Rule:                           

(4   )(4   )= 4       =4               

3.Inverse Rule:    

((5x   y      ) )   = 1/ 25x    y  

                = y     /25x  

5.Power of a product:

((4)(5))   = (4   )(5   )

7. Power of a quotient:

()

2. Quotient Rule:             

(7    )/(7   )= 7        = 7

4. Zero Rule:

    27   = 1

6.Power of a power:

(8^3)^2= (8)^(3)(2)= 8^(6)

     5     3       5+3      8                                10      5       10-5       5

n

     2    -8   2                4    -16                         0

                     16        4

 2            2     2

Examples:

Square Root:

12= 10+ √ 2b-10 (Subtract 10 on both sides)

(2) =(√ 2b-10) (Square both sides)

4= 2b-10 (Add 10 on both sides)

14=2b (Divide 2 on both sides )

7=b (Final answer)   

Cube Root:

15=11+√ 3b+10 (Subtract 11 on both sides)

(4) =(√ 3b+10) (Cube both sides)

64= 3b+10 (Subtract 10 on both sides)

54=3b (Divide by three on both sides)

18=b (Final answer)

(-10-2x)    = -125

((-10-2x)   )    =(-125) 

-10-2x=25

-2x=35

x=-17.5 or -35/2

6p-2    =p+1

6p-2=p   +2p+1

p   -4p+3

(p-3)(p-1)

p=3 and 1

3

2

_

3

2

2

3

2

3

1

2

2

2

Graphing Radical Functions

First, you have to look at your equation. 

For example:

y=

       x+6+3

Remember what is added or subtracted to x is your x 

coordinate. Also, whatever your adding or subtracting 

is opposite. The number outside the radical is your y 

coordinate and that number stays the same.

x+6

The shape of a square root is a parabola on its 

side but only the positive side. This is do to 

that a square root has 2 answers but a 

function can only use one.

The shape of a cube root is an s shape on its 

side. Because a cube root has one answer no 

matter if its negative or positive so there sare 

no limits on its domain.

To start graphing a square root start with the 

the vertex from your equation. y=    x-6   +3

which is (6,3), start with your vertex and 

follow the pattern of 1,3,5, so it up 1 over 1, 

up 1 over 3, etc until you run out of graph

3

To graph a cube root you star with the vertex 

of the equation, 

y=     x-6    +3, which is still (6,3). Then you 

follow the pattern of 1, 3, 7 both for the 

positive and negative. so it up 1 over 1, up 1 

over 3. Then you start from the vertex and do 

the opposite

And finally to graph a stretch graph is the 

same way like the other 2, the only difference 

is the number of spaces you go up or down.

y=2     x-6   +3. Your vertx is (6,3)

next you go up and over but this time instead 

of 1   up over 1, 1 up over , it will be up 2 over 

1, up 2 over 3... etc

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