# Thread: How to simplify velocity magnitude equations?

1. ## How to simplify velocity magnitude equations?

In my calculus class a few days ago, my professor was going over how to find three dimensional velocity and its magnitude. In the problem, the magnitude of the velocity came out to be $\displaystyle \sqrt{9+36t^2+36t^4}=3+6t^2$.

My question is, how did he simplify the equation under the square root so that it became $\displaystyle 3+6t^2$?

For the homework, I need to simplify these two square root equations to get something without a square root sign (if possible). Can you guys help me out with this? Thanks.
1) $\displaystyle \sqrt{9t^4+36+36t^2}$

2) $\displaystyle \sqrt{(-3cost)^2+(-4cost)^2+(-5sint)^2)}$

2. Hello Infernorage!

Originally Posted by Infernorage
In my calculus class a few days ago, my professor was going over how to find three dimensional velocity and its magnitude. In the problem, the magnitude of the velocity came out to be $\displaystyle \sqrt{9+36t^2+36t^4}=3+6t^2$.
Do you know this rule:

$\displaystyle (a+b)^2 = a^2+2ab+b^2$

It is not difficult to see that $\displaystyle 9+36t^2+36t^4 = (6t^2+3)^2$

So

$\displaystyle \sqrt{9+36t^2+36t^4}=3+6t^2 = \sqrt{(6t^2+3)^2} = 3+6t^2$

Originally Posted by Infernorage
My question is, how did he simplify the equation under the square root so that it became $\displaystyle 3+6t^2$?

For the homework, I need to simplify these two square root equations to get something without a square root sign (if possible). Can you guys help me out with this? Thanks.
1) $\displaystyle \sqrt{9t^4+36+36t^2}$
This one is tricky.
I told you $\displaystyle (a+b)^2 = a^2+2ab+b^2$. You can't use it here unless :

$\displaystyle \sqrt{9t^4+36+36t^2} = \sqrt{9*(t^4+4+4t^2)}$

Hence

$\displaystyle \sqrt{9*(t^4+4+4t^2)} = \sqrt{9(t^2+2)^2}$

Originally Posted by Infernorage
2) $\displaystyle \sqrt{(-3cost)^2+(-4cost)^2+(-5sint)^2)}$
It is $\displaystyle sin^2+cos^2=1$

and furthermore

$\displaystyle \sqrt{(-3cost)^2+(-4cost)^2+(-5sint)^2)}$

$\displaystyle =\sqrt{9cos^2t+16cos^2t+25sin^2t}$

$\displaystyle =\sqrt{25cos^2t+25sin^2t}$

$\displaystyle =\sqrt{25(cos^2t+sin^2t)}$

$\displaystyle =\sqrt{25*(1)} = \sqrt{25}$

Do you understand?

Yours
Rapha