I need some help with these problems please?

Question

divide (x^2-36y^2) over (7x^2-42y) divided by (x^2+6xy)

it looks like this
x^2-36y^2
________ by (x^2+6xy
7x^2-42y

and the other problem is (2x-6) 0ver (21) divided by (5x-15) over (12)
this one looks like this
2x-6...... ........5x-15
______by_______
21......... .......12.
the dots are just to spererate the numbers so please ignore thoes.

if there is anyone who can help me I will be thankful for it.

Answer 1

(1) [ (x^2 - 36y^2) / (7x^2 - 42y) ] / [x^2 + 6xy]

Factor each bracketed expression.

[ (x - 6y)(x + 6y) / (7(x^2 - 6y)) ] / [x(x + 6y) ]

To get rid of this complex fraction, multiply top and bottom (of the outer fraction) by 7(x^2 - 6y).

[ (x - 6y)(x + 6y) ] / [ x(x + 6y) (7) (x^2 - 6y) ]

Note the cancellation now.

(x - 6y) / [x(7)(x^2 - 6y)]

(x - 6y) / [7x (x^2 - 6y) ]

2)

[ (2x - 6)/21 ] / [ (5x - 15)/12 ]

To solve this, multiply top and bottom by the lowest common denominator of 12 and 21. Note that
12 = 2 x 2 x 3
21 = 7 x 3, so by the elementary school rules of finding LCM,
the LCM should be 2 x 2 x 3 x 7 = 84.

Multiply top and bottom by 84, to get

84[ (2x - 6)/21 ] / 84[ (5x - 15)/12 ]

[4(2x - 6)] / [7(5x - 15)]

Next, factor each term in parentheses.

[4(2)(x - 3)] / [7(5)(x - 3)]

Now, cancel (x - 3)

[4(2)] / [7(5)]

8/35

Answer 2

you might wanna check and see if you've mis-typed your first question 'cause i've got a strong feeling that you have, but i shall try my best to solve it anyway :)

hopefully you know this: something divided by (a/b) means it's multiplied by (b/a). here's an example: picture an elephant (any colour you like, preferrably gray), now that elephant says "divide me by 3/5", what he actually means is he wants you to multiply him by 5/3, get it?

mathematically:
n / (a/b) = n * (b/a)

1.
(x^2 - 36y^2) / (7x^2 - 42y) divided by (x^2 + 6xy)
= (x^2 - 36y^2) / (7x^2 - 42y) multiplied by 1 / (x^2 + 6xy)

now i'll split this into 2 parts: numerators and denominators

multiplying the numerators:
(x^2 - 36y^2) * 1
= (x^2 - 36y^2)
= (x + 6y) (x - 6y) [formula: x^2 - y^2 = (x+y) (x-y)]

multiplying the denominators:
(7x^2 - 42y) * (x^2 + 6xy)
= (7) (x^2 - 6y) * (x) (x + 6y) [basic factorization]
= 7x (x^2 - 6y) (x + 6y)

observe that the (x + 6y) cancel each other out, so your final answer is:
(x - 6y) over (7x) (x^2 - 6y)

2.
(2x - 6) / (21) divided by (5x-15) / (12)
= (2x - 6) / (21) multiplied by (12) / (5x-15)

multiplying numerators:
(2x - 6) * 12
= 2 (x - 3) * 12 [basic factorization]
= 24 (x - 3)

multiplying denominators:
21 * (5x - 15)
= 21 * 5 (x - 3)
= 105 (x - 3)

observe that (x - 3) cancel each other out and (24/105) can be reduced by dividing by a common factor of 3, so the final answer is:
8/35

Answer 3

x^2-36y^2
________ by (x^2+6xy
7x^2-42y

= [(x^2-36y^2 ) / (7x^2-42y )] / (x^2+6xy )

= [(x-6y)(x+6y)] / [ 6(x^2-6)x(x+6y) ]
= (x-6y) / [6x(x^2-6) ]


2x-6...... ........5x-15
______by_______
21......... .......12.
= [ (2x-6)/21 ] / [(5x-15)/12 ]
= [ (2x-6)*12 ] / [ 21*(5x-15) ]
= 8(x-3) / [35(x-3) ]
= 8/35