help please... how do i simplify this problem?

Question

4 √72 + 3 √18 - 5 √6



3x^2 y^2
-------------
6xy^2

please show the steps.

Answer 1

4 √72 + 3 √18 - 5 √6
4 √(9*4*2) + 3 √(9*2) - 5 √6
24√2 + 9√2 - 5√6
33√2 - 5√6

(3x^2 y^2)/(6xy^2)
cancel out the two y^2
3/6 reduces to 1/2
x^2/x = x
so the answer is
x/2

Answer 2

4 √72 + 3 √18 - 5 √6
the first step is to find the common factors of the the things that are having the square roots.
72 is 9*8
or 3*3*2*2*2
or 6*6*2
or (6^2)*2
a different expresion √72
that has the identical value is (6*√2 )

In a similar fashion √18
can be rewriten as (3*√2)

One simplification then would be to substitue these equivalent forms and see what terms can be combined.

4 * (6*√2 )+ 3 *(3*√2)- 5 √6
24*√2+9*√2-5*√6
33*√2-5*√6
I believe this to be the simplest form but ther is another comparable form
(33-5*√3)*√2

Each of these forms have two different invocations of the square root function two multiplications and one subtraction.

3x^2 y^2
-------------
6xy^2


The numerator and denominator have several common factors.
the constant in the numerator and the constant in the denominator are both divisible by 3
If you divide both the numerator and the denominator of a fraction by the the same number then the resultant fraction has the same value as the original fraction.
first step then is the eliminate the 3 factor from both the numerator and the denominator.

x^2*y^2/2*x*y^2

Both the numerator and denominator contain y^2
second step is to eliminate the y^2 factor from both.

x^2/2*x

there is an x factor in both to be eliminated by dividing both numerator and denominator by x

The result that is called the simplest form is thus
x/2

Answer 3

4√72 + 3√18 - 5√6

First, factor the terms in the radicals. You're looking to extract all perfect square factors:

4√(2*36) + 3√(2*9) - 5√6

Next, extract the perfect square factors, and simplify:

4√2√36 + 3√2√9 - 5√6
4*6√2 + 3*3√2 - 5√6
24√2 + 9√2 - 5√6

Next, combine like terms:

33√2 - 5√6

And we are done, although note that we could factor this further as (33-5√3)√2, but this would not make it simpler.

For the second one, note that the 3, the y², and one of the x's cancel, leaving you with:

x/2

Answer 4

4 √72 + 3 √18 - 5 √6
4 √(36*2) + 3 √(9*2) - 5 √6
4*6√2 + 3*3√2 - 5√6
24√2 + 9√2 - 5√6
33√2 - 5√6


3x^2 y^2
-------------
6xy^2

is

3*x*x*y*y
-------------
2*3*x*y*y

cancel out what is common and you get

x
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2