2TH §4.3 Multiplying square roots

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§4.3 Multiplying square roots

is on the menu today!

However, first there follows a recap on 

adding and subtracting roots!

Write as short as possible

y=3√​x​​​−6√​2​​​+6√​x​​​−2√​2​​​

1. y=9√​x​​​−4√​2​​​
2. y=9√​x​​​−8√​2​​​
3. y=−3√​x​​​+4√​2​​​
4. Cannot be written any shorter.

Which of these does not exist?

1. √​25​​​
2. −√​25​​​
3. √​−25​​​
4. they all exist

Calculate without a calculator

(√​7​​​)​2​​=

Recap

- You cannot take the square root of a negative number

So                               is impossible

- You can only add and subtract square roots with the same number or variable underneath the square root sign. Like:

      

- If you square a square root, you get the number underneath the square root sign (if the square root exists). So for example:

                                   = 7        (this last item is NEW; it's from §4.3 !)

   

√​−6​​​

√​5​​​+3√​5​​​=4√​5​​​

(√​7​[?]​​​​​)​2​​

Multiplying square roots

We've seen that

because: 

 area of the square =               x            =               = 5

As a rule we say:

√​5​​​

√​5​​​

√​25​​​

Let's try to find out if you can multiply roots:

Is this true:

               x                  =                   ?

What does your intuition tell you about this?

ANS

√​4​​​

√​9​​​

√​36​​​

root 9 x root 4 = root 36

1. this is correct
2. not correct

Here you see WHY you can multiply roots:

It is correct to do:

               x                  =                    and this is why:

       2      x      3         =           6 !

As you see, if you rewrite the roots as numbers, the multiplication is correct!

√​4​​​

√​9​​​

√​36​​​

Multiplying square roots

If you multiply different square roots, the following rule applies:

                                                          (if both square roots exist)

Examples:

(

root 3 x root 7 =

1. square root 21
2. square root 10
3. 21

root 18 x root 2 = .....

1. 6
2. square root 36
3. square root 20

explanation:

                                             x                      =                 

This is correct, as you may multiply roots!

However, if possible we have to SIMPLIFY, so the best answer is:

                          = 6 (!), because 6 x 6 = 36

√​18​​​

√​2​​​

√​36​​​

√​36​​​

Homework 

Do §4.3 Exercises S18, 20, S21, 22, 23 and 24

Check your answers with the solutions book!

Find it easy? Try the Challenging exercises C5 and C6

Multiplying square roots

If you multiply square root numbers, the following rule applies:

                                                                         (if both square roots exist)          

Examples:     (take note: in the 2nd one there is NO subtraction!)