Question

It is crucial that I understand what you are doing to solve this so please explain this step by step.

Simplify.
https://media.glynlyon.com/g_alg01_2013/8/test13.gif

Answer 1

This is the main "Type" of problem I am really confused with.

Answer 2

\[\frac{8\sqrt{6mn}+6\sqrt{8mn}}{2\sqrt{2mn}}\]One thing we do when there is a radical in the denominator is "rationalize" by multiplying both the numerator and denominator by the radical.

In this case, we would multiply the fraction by \[\frac{\sqrt{2mn}}{\sqrt{2mn}}\]which doesn't change the value of the fraction at all (that reduces to 1, after all), but turns it into a form where we can simplify it some more. Why don't you try doing that and see how far you get?

Answer 3

\[\frac{ 8\sqrt{12mn}+12\sqrt{16mn} }{ 4\sqrt{4mn} }\] ?

Answer 4

I'm sorry I suck at this, I must be frustrating for you haha :\

Answer 5

not quite, but a good try!

maybe a bit of refresher is in order.

\[\sqrt{a}*\sqrt{b} = \sqrt{a*b}\]

looking at only the denominator:
\[2\sqrt{2mn} * \sqrt{2mn} = \]
?

Answer 6

Remember also that \[\sqrt{a*a} = a\]

Answer 7

ohhh so the square root signs would go away?

Answer 8

Yes, \[2\sqrt{2mn}*\sqrt{2mn} = 2\sqrt{2*2*m*m*n*n} = 2*(2*m*n) = 4mn\]

Answer 9

so it'd be \[2 \times 2mn\]?

Answer 10

Right. Or 4mn.

Now, the top, when we multiply that by the same thing, what do we get?
\[(8\sqrt{6mn} + 6\sqrt{8mn})*\sqrt{2mn} = \]

Answer 11

\[48mn+48mn?\]

Answer 12

no. let's break it down, just do the left term:
\[8\sqrt{6mn}*\sqrt{2mn} = 8\sqrt{6*2*m*m*n*n} = \]

Answer 13

hint: 6 = 3*2

Answer 14

oh, I was doing it as if it were two.. so it'd be,\[3\sqrt{2mn}\]

Answer 15

no...
\[8\sqrt{6mn}*\sqrt{2mn} = 8\sqrt{6*2*m*m*n*n} = 8\sqrt{3*2*2*m*m*n*n}\]Each time we have a pair of identical factors under the radical sign, we can remove them and replace one of them outside the radical sign:
\[\sqrt{16} = \sqrt{4*4} = 4\]\[\sqrt{12} = \sqrt{2*2*3} = 2\sqrt{3}\]

Answer 16

\[8\sqrt{3*2*2*m*m*n*n} =\]

Answer 17

\[8\sqrt{6mn}\]?

Answer 18

:-(

\[8\sqrt{3*(2*2)*(m*m)*(n*n)} = 8\sqrt{3}*\sqrt{2*2}*\sqrt{m*m}*\sqrt{n*n} = \]

Answer 19

remember, \[\sqrt{a*a} = a\]

Answer 20

so, 2*2 would be just 2 and then m*m=m and then n*n=n right?
and 3 would stay in the square root because it doesn't double

Answer 21

right, so the whole enchilada would be?

Answer 22

Lol the whole enchilada... *hem hem*... \[2mn \sqrt{3} ?\]

Answer 23

what happened to that 8 we had lurking about?

Answer 24

oh, you add that to the 2mn?

Answer 25

well, not add, multiply

Answer 26

ohhhh ok so it'd be \[16mn \sqrt{3}\]?

Answer 27

yes, though it would be more typical to write it \[16\sqrt{3} m n \]

Answer 28

Now, how about the other half of the numerator?
\[6\sqrt{8 mn} * \sqrt{2mn} = \]

Answer 29

ok...so then the other side would be \[3\sqrt{16mn}\]

Answer 30

try again please, I'll pretend you didn't just crush my soul :-)

Answer 31

lol ok... ok, one sec... minute

Answer 32

remember, \[\sqrt{a}*\sqrt{a} = \sqrt{a*a} = a\]

Answer 33

\[12\sqrt{4nm}?\]

Answer 34

it's either 4 or 8 on the inside...

Answer 35

getting better, but still not correct.

\[6\sqrt{8mn}*\sqrt{2mn} = 6\sqrt{8*2*m*m*n*n}=\]

Answer 36

and 8 is 2*4 so \[6\sqrt{4 *2 (2*2)(m*m)(n*n)}\]

Answer 37

right?

Answer 38

achingly close, yet incorrect!

\[6\sqrt{8*2*m*m*n*n} = 6\sqrt{(2*2*2)*2*(m*m)*(n*n)} =\]

Answer 39

(you stuck in an extra 2)

Answer 40

well, if you factor it like that, it's easier to spot what you can pull out, if you haven't trained yourself to see "8*2=16 which is the same as 4*4 so I can just pull out a 4"

Answer 41

I'm taking baby steps so that everything is as clear as possible. with practice, you can skip a lot of the steps and still even get the right answer sometimes :-)

Answer 42

I would look at \[6\sqrt{8mn}*\sqrt{2mn}\] and immediately think \(24mn\) because the two pairs of \(mn\) under the square root sign become one outside the square root sign, and \(8*2=16=4*4\) under the square root sign becomes 4 outside of it, and so I'm left with \[6*4*mn=24mn\]

Answer 43

You'll get better with practice!

Answer 44

oh ok... Thank you for being patient and talking me through the problem, I've been confused with radicals (as you know ) for a while.

Answer 45

yeah, sometimes it takes a while, and then suddenly it clicks.

Answer 46

so is it \[16\sqrt{4mn}\]?

Answer 47

well, \sqrt{4} = 2, right? so there would be no reason to leave it under the radical sign. and we multiplied \(\sqrt{mn}*\sqrt{mn}\) (they both had some other stuff, but this is the part I'm concerned with at the moment), so we should have \(mn\), not \(\sqrt{mn}\)

Answer 48

that should have started "well, \(\sqrt{4} = 2\)"

Answer 49

mn and not \[\sqrt{mn}\] because it was doubled?

Answer 50

because there were two of each, yes

Answer 51

Say, do you know about exponents yet? I wonder if perhaps writing this in a different way would make it clearer for you...

Answer 52

yep I do :)

Answer 53

okay, \[\sqrt{x} = x^{\frac{1}{2}}\]Did you know that?

Answer 54

so, we could write \[6\sqrt{8mn} = 6*8^{\frac{1}2}*m^{\frac{1}{2}}*n^{\frac{1}{2}}\]and if we multiply that by \[\sqrt{2mn} = 2^{\frac{1}{2}}*m^{\frac{1}{2}}*n^{\frac{1}{2}}\] we have\[ 6*8^{\frac{1}2}*m^{\frac{1}{2}}*n^{\frac{1}{2}}*2^{\frac{1}{2}}*m^{\frac{1}{2}}*n^{\frac{1}{2}} = 6*8^{\frac{1}{2}}*2^{\frac{1}{2}}*m^{\frac{1}{2}+\frac{1}{2}}*n^{\frac{1}{2}+\frac{1}{2}}\]\[=6*8^{\frac{1}{2}}*2^{\frac{1}{2}}*m^{1}*n^{1}=6*(8*2)^{\frac{1}{2}}*m*n = 6*(16)^{\frac{1}{2}}*mn\]\[ = 6*\sqrt{16}*mn = 6*4*mn=24mn\]

Answer 55

anyhow, the two terms we got for the numerator are

\[16\sqrt{3}mn + 24mn\]and the denominator was \(4mn\) so our overall fraction is (drum roll please....)
\[\frac{16\sqrt{3}mn + 24mn}{4mn}\]

Now, do you see any common factors you can cancel out of that?

Answer 56

Looks like everything has \(mn\) in it, right?

\[\frac{16\sqrt{3}\cancel{mn}+24\cancel{mn}}{4\cancel{mn}} = \frac{16\sqrt{3}+24}{4} \]See any more common factors we can take out?

Answer 57

looks like we can factor a 4 out of the top as well:
\[\frac{4*(4\sqrt{3}+6)}{4} = \]

Answer 58

What do you get after you cancel the common factor of 4 from both numerator and denominator?

Answer 59

Something that looks a whole lot simpler than what we started with :-)

Answer 60

Sorry I left for a while, yep haha Thank you so much, I think i get it now finally :) haha