How to simplify $42\sqrt{45} \over 7\sqrt{35}$?

Question

The problem is $$42\sqrt{45} \over 7\sqrt{35}$$ HELP! My daughter's math sheet shows how to reduce the squareroots, but the examples all use the same square-root; the problems show two different numbers square-rooted. Can you please help work this problem so I know how to help her with this assignment?

Thanks!

@MaureenMurphy: I've edited your formula to help us understand it better. Please check and make sure I formatted the formula correctly and that it looks like the one in the homework sheet. If it doesn't, tell me what is wrong and I will correct it. — MJD, Feb 28 '13 at 22:09
@Dominic: Of course she doesn't mean that. And this person probably hasn't even heard of $\LaTeX$, so be gentle. — MJD, Feb 28 '13 at 22:10
@Mjd i didn't downvote i just want to know what the question is, i can't unterstand the downvotes — Dominic Michaelis, Feb 28 '13 at 22:11
Know the numbers out front are not little they are big, sorry, no idea how to make a sqrt symbol on here, darn technology! — Maureen Murphy, Feb 28 '13 at 22:11
@Dominic I'm not criticizing your vote, which I don't know anything about; I'm criticizing your comment. — MJD, Feb 28 '13 at 22:11
@MJD you are right I have absolutely no idea what latex means in this realm, in my realm they are gloves that I put on before performing Patient care — Maureen Murphy, Feb 28 '13 at 22:13

score 5 · Answer 1 · answered Feb 28 '13 at 22:11

5

Note that $45=5\cdot 9$, hence $\sqrt{45}=3\cdot \sqrt 5$. Also $\sqrt{35}=\sqrt 5\cdot\sqrt 7$. That should help simplify a lot.

answered Feb 28 '13 at 22:11

Hagen von Eitzen

374,180

score 5 · Answer 2 · answered Feb 28 '13 at 22:13

5

Remember that for positive numbers $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$. So we can simplify as follows:

$$\frac{42\times\sqrt{45}}{7\times\sqrt{35}}=\frac{6\times7\times\sqrt9\times\sqrt5}{7\times\sqrt7\times\sqrt5}=\frac{6\times 3}{\sqrt7}=\frac{18}{\sqrt7}=\frac{18}{\sqrt7}\times1=\frac{18}{\sqrt7}\times\frac{\sqrt7}{\sqrt7}=\frac{18\times\sqrt7}{7}$$

answered Feb 28 '13 at 22:13

Asaf Karagila

393,674

3

She says can't end with a radical on the bottom? – Maureen Murphy Feb 28 '13 at 22:16
1

Then $$\frac{18}{\sqrt 7}=\frac{18\sqrt 7}{7}\ldots$$ – DonAntonio Feb 28 '13 at 22:17
10

History teaches us that radicals always try to get to the top. Mathematics is no exception! – Asaf Karagila Feb 28 '13 at 22:19
Please continue we're writing it all down.... – Maureen Murphy Feb 28 '13 at 22:19
More? There isn't any simpler form. You can maybe insert the $18$ into the radical on top. Or do you mean fake history lessons (which are not fake at all)? – Asaf Karagila Feb 28 '13 at 22:21

amWhy · Accepted Answer · 2013-02-28T22:27:43.627

2

$$\frac{42\sqrt{45}}{7\sqrt{35}} = \frac{6\times 7\sqrt{9\times 5}}{7\sqrt{7\times 5}} $$ $$ = \frac{6\sqrt 9 \times \sqrt 5}{\sqrt 7 \times \sqrt 5} = \frac{6\times \sqrt 9}{\sqrt 7}$$ $$= \frac{6 \times 3}{\sqrt 7} \times \frac{\sqrt 7}{\sqrt 7} = \frac{18\sqrt 7}{ 7}$$

Cancel common factors, then multiply denominator to clear the square root.

edited Feb 28 '13 at 22:27

answered Feb 28 '13 at 22:14

amWhy

209,954

You forgot to take the $7$ from the denominator when you divided the $42$ into $6$. – Asaf Karagila Feb 28 '13 at 22:20
Thank you all so much for the step by step, she gets it now, still as clear as mud to me!!!! LOL – Maureen Murphy Feb 28 '13 at 22:28
I'll stick to Nursing, am definately not a mathmetician – Maureen Murphy Feb 28 '13 at 22:29
Just click on the checkmark $\checkmark$ to the left of the answer you'd like to accept! – amWhy Feb 28 '13 at 22:32
Thank you all for the help, such a great group! – Maureen Murphy Feb 28 '13 at 22:34
3

You're welcome, and I'm sure we all feel the same way. That's why we're here: we don't get paid, so we're here for the sheer pleasure of helping when we can! – amWhy Feb 28 '13 at 22:36
Kill the fractions Amy. Just do it! ;-) + – Mikasa Mar 01 '13 at 06:34

score 1 · Answer 4 · answered Feb 28 '13 at 22:24

$$\frac{42 \sqrt{45}}{7 \sqrt{35}} \;\; = \;\; \frac{42}{7} \cdot \frac{\sqrt{45}}{\sqrt{35}} \;\; = \;\; \frac{42}{7} \cdot \sqrt{\frac{45}{35}} \;\; = \;\; \frac{6}{1} \cdot \sqrt{\frac{9}{7}}$$

$$ = \;\; \frac{6}{1} \cdot \frac{\sqrt{9}}{\sqrt{7}} \;\; = \;\; \frac{6}{1} \cdot \frac{3}{\sqrt{7}} \;\; = \;\; \frac{6}{1} \cdot \frac{3}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$$

$$= \;\; \frac{6 \cdot 3 \cdot \sqrt{7}}{1 \cdot \sqrt{7} \cdot \sqrt{7}} \;\; =\;\; \frac{18\sqrt{7}}{7}$$

How to simplify $42\sqrt{45} \over 7\sqrt{35}$?

4 Answers4