Wrong conclusion two basic surd simplifications.

[localclown]

Hi, first timer and looking to improve my math skills. Thanks in advance for any assistance provided.

[MATH]\text {Simplify the following surd}[/MATH][MATH]\frac{\sqrt{88}}{2}[/MATH]
I worked this out as:

[MATH]\sqrt{88}\div{2}={88}\div{2}={44}\text{ so }{4}\times{11}={2}\sqrt{11}[/MATH]
However the correct answer is [MATH]\sqrt{22}[/MATH]. Please can you explain how where I went wrong in coming to the correct conclusion?

And:

[MATH]\text {Simplify the following surd}[/MATH][MATH]\frac{14\sqrt{30}}{16\sqrt{3}}[/MATH]
I clumsily fumbled through this not knowing what I was doing:

[MATH]{14}\div{16}={0.875}[/MATH][MATH]\sqrt{30}\div\sqrt{3}=\sqrt{10}[/MATH][MATH]={0.875}\sqrt{10}[/MATH]
When the correct answer was:

[MATH]\frac{7\sqrt{10}}{8}[/MATH]
EDIT 14:54: So I noticed in the previous question the whole numbers, 14 and 16 have been halved. Why is this? I'm not making the connection between addressing the whole numbers first, and the square roots after but somehow reaching 14/2 and 16/2.

EDIT 15:00: Thanks to MarkFL for providing the scripts on how to correctly display these fractions.

Thanks!

 

[MarkFL]

Hello, and welcome to FMH!

[MATH]\frac{\sqrt{88}}{2}=\frac{\sqrt{4\cdot22}}{2}=\frac{2\sqrt{22}}{2}=\sqrt{22}[/MATH]
[MATH]\frac{14\sqrt{30}}{16\sqrt{3}}=\frac{7\sqrt{10}}{8}[/MATH]
You can quote my post to see how I wrote the fractions.

 

[localclown]

Hello, and welcome to FMH!

[MATH]\frac{\sqrt{88}}{2}=\frac{\sqrt{4\cdot22}}{2}=\frac{2\sqrt{22}}{2}=\sqrt{22}[/MATH]
[MATH]\frac{14\sqrt{30}}{16\sqrt{3}}=\frac{7\sqrt{10}}{8}[/MATH]
You can quote my post to see how I wrote the fractions.

Thank you for the information.

The error was whilst I recognized that the GCF of 88 was 4, I proceeded to:

[MATH]\frac{\sqrt{88}}{2}={88}\div{2}[/MATH]
This meant I treated the surd as a whole number rather than what it is, a surd!

I now understand that:

[MATH]\frac{\sqrt{88}}{2}=\frac{\sqrt{4\times22}}{2}[/MATH]

Is the correct equation!

However I am still confused with regards to the latter simplification. I will take another look at it.

EDIT 15:49: Do we find 7 and 8 because the GCF of 14 and 16 is 2? And therefore:

[MATH]\frac{14\sqrt{30}}{16\sqrt{3}}=\frac{14}{2}={7}[/MATH]
[MATH]\frac{14\sqrt{30}}{16\sqrt{3}}=\frac{16}{2}={8}[/MATH]
[MATH]\frac{14\sqrt{30}}{16\sqrt{3}}=\frac{\sqrt{30}}{\sqrt{3}}=\sqrt{10}[/MATH]
So:

[MATH]\frac{14\sqrt{30}}{16\sqrt{3}}=\frac{7\sqrt{10}}{8}[/MATH]
I think I am closer to understanding this question. However, why is it that we find [MATH]\sqrt{10}[/MATH] at the top rather than the bottom? Are they interchangeable?

Many thanks!

 

[MarkFL]

Yes, both 14 and 16 have 2 as a factor, and we can divide it out.

By the way, when you obtained:

[MATH]0.875\sqrt{10}[/MATH]
This isn't incorrect, it's just not in the expected form.

 

[localclown]

Yes, both 14 and 16 have 2 as a factor, and we can divide it out.

By the way, when you obtained:

[MATH]0.875\sqrt{10}[/MATH]
This isn't incorrect, it's just not in the expected form.

Mark you've been so helpful, thank you! If you have another moment could you please look at what I edited above at 15:49, just a moment or two subsequent to the quoted post? Very much appreciated!

 

[MarkFL]

Your work at 15:49 isn't correct. If I were to include more steps, I would write:

[MATH]\frac{14\sqrt{30}}{16\sqrt{3}}=\frac{2\cdot7\sqrt{3\cdot10}}{2\cdot8\sqrt{3}}=\frac{2\sqrt{3}}{2\sqrt{3}}\cdot\frac{7\sqrt{10}}{8}=\frac{7\sqrt{10}}{8}[/MATH]

 

[HallsofIvy]

Thank you for the information.

The error was whilst I recognized that the GCF of 88 was 4, I proceeded to:

[MATH]\frac{\sqrt{88}}{2}={88}\div{2}[/MATH]
This meant I treated the surd as a whole number rather than what it is, a surd!

I now understand that:

[MATH]\frac{\sqrt{88}}{2}=\frac{\sqrt{4\times22}}{2}[/MATH]

Is the correct equation!

However I am still confused with regards to the latter simplification. I will take another look at it.

EDIT 15:49: Do we find 7 and 8 because the GCF of 14 and 16 is 2? And therefore:

[MATH]\frac{14\sqrt{30}}{16\sqrt{3}}=\frac{14}{2}={7}[/MATH]
[MATH]\frac{14\sqrt{30}}{16\sqrt{3}}=\frac{16}{2}={8}[/MATH]

I really hate the way you are writing these and are using the "=" sign so loosely!
Do you really think that [math]\frac{14\sqrt{30}}{16\sqrt{3}}[/math] is equal to both 7 and 8? That is what you have written!

No, [math]\frac{14\sqrt{30}}{16\sqrt{3}}[/math] is not equal to [math]\frac{14}{2}[/math]! Where did you get such an idea?