Question

What is the geometric mean of 12 and 18?
a. 1.5
b. √6
c. 15
d. 6√6
Also, if you could explain how to find a geometric mean, that would be awesome!
Thanks! x

Answer 1

\(\bf \cfrac{12}{{\color{red}{ x}}}=\cfrac{{\color{red}{ x}}}{18}\qquad {\color{red}{ \textit{geometric mean}}}\)

Answer 2

solve for "x"

Answer 3

I got x=108.

Answer 4

108? hmm how did you get 108?

Answer 5

I cross multiplied? It's been a while since I've had to do problems like this, so I am probably doing it wrong….

Answer 6

well, cross-multiplying is ... correct.. so \(\bf \cfrac{12}{{\color{red}{ x}}}=\cfrac{{\color{red}{ x}}}{18}\implies 12\cdot 18={\color{red}{ x\cdot x}}\implies 216=x^2\\ \quad \\
\textit{taking }\sqrt{\qquad }\textit{ to both sides}\\ \quad \\
\sqrt{216}=\sqrt{x^2}\implies \sqrt{216}=x\\ \quad \\
{\color{blue}{ 216\implies 2\cdot 2\cdot 2\cdot 3\cdot 3\cdot 3\implies 2\cdot 2^2\cdot 3\cdot 3^2}}\qquad thus\\ \quad \\
\sqrt{216}=x\implies \sqrt{2\cdot 2^2\cdot 3\cdot 3^2}=x\implies

\large \sqrt[{\color{red}{ 2}}]{2\cdot 2^{\color{red}{ 2}}\cdot 3\cdot 3^{\color{red}{ 2}}}=x\)

Answer 7

take what you can from the radical, and leave the rest :)

Answer 8

Ah, that's what I did wrong! I said 2x, not x squared. This is amazing, thank you so much! So, when I did the square root of 216, I got 14.696938… Would I just round up to 15?

Answer 9

well, is not one of your choices for one, though is correct decimally

Answer 10

notice the radicand, recall that we take out from the radical what MATCHES the root

Answer 11

Ok, I guess I don't know how to do that, although I know what you're saying. Thank you so so much, by the way.

Answer 12

one sec

Answer 13

\(\Large \sqrt[{\color{red}{ 2}}]{2\cdot {\color{blue}{ 2}}^{\color{red}{ 2}}\cdot 3\cdot {\color{blue}{ 3}}^{\color{red}{ 2}}}=x\implies {\color{blue}{ 2\cdot 3}}\sqrt{2\cdot 3}\)

Answer 14

you take out anything that matches the "root", thus

Answer 15

Ohh… Thank you sooo much!! You've been so much help, I really appreciate it! One last question, if you don't mind:) I'm working on the same type of question in my assignment:
What is the geometric mean of 2 and 36. So I got to finding the square root of 70. How do I break it down like you did to the square root of 216?

Answer 16

70? where did you get 70 from?

Answer 17

Oh oops, I meant 72

Answer 18

got a calculator?

Answer 19

yes

Answer 20

ok... try dividing 72 by 2, if you can, again by 2, and on
if you can't try some other small number, like 3 and on

Answer 21

what factors would that give you?

Answer 22

Ok, so it would be broken down into 2·2·2·3·3

Answer 23

\(\bf \cfrac{2}{x}=\cfrac{x}{36}\implies 72=x^2\implies \sqrt{72}=x\\ \quad \\
{\color{blue}{ 72\implies 2\cdot 2\cdot 2\cdot 3\cdot 3\implies 2\cdot 2^2\cdot 3^2}}\\ \quad \\
\sqrt{72}=x\implies \sqrt{2\cdot 2^2\cdot 3^2}=x\)

Answer 24

\(\bf \sqrt{72}=x\implies \sqrt{2\cdot 2^2\cdot 3^2}=x\implies 2\cdot 3\sqrt{2}=x\implies 6\sqrt{2}=x\)

Answer 25

Ok, that's what I got. So I would have 6√2?

Answer 26

Awesome. Thank you so much again!

Answer 27

yw