# Thread: Simple Simplification

1. ## Simple Simplification

Hey,

Anyone know if this can be simplified any further?

$\displaystyle [\sqrt{4(cos^2t) + 9(sin^2t)} ]^3$

Cheers

2. Hi,
Originally Posted by tsal15
Anyone know if this can be simplified any further?

$\displaystyle [\sqrt{4(cos^2t) + 9(sin^2t)} ]^3$
Yes, it can be simplified : $\displaystyle \left[\sqrt{4\cos^2t + 9\sin^2t} \right]^3=\left[\sqrt{4(\cos^2t +\sin^2t)+ 5\sin^2t} \right]^3$ and remember that $\displaystyle \cos^2t+\sin^2t=1$.

3. Originally Posted by flyingsquirrel
Hi,

Yes, it can be simplified : $\displaystyle \left[\sqrt{4\cos^2t + 9\sin^2t} \right]^3=\left[\sqrt{4(\cos^2t +\sin^2t)+ 5\sin^2t} \right]^3$ and remember that $\displaystyle \cos^2t+\sin^2t=1$.
hey flyingsquirrel,

Is this as far as I can go?

mathematically speaking...

$\displaystyle \left[\sqrt{5 + 5sin^2t} \right]^3$

4. Originally Posted by tsal15
Is this as far as I can go?

mathematically speaking...

$\displaystyle \left[\sqrt{5 + 5sin^2t} \right]^3$
Mathematically speaking, you're going too far.

$\displaystyle \left[\sqrt{4\cos^2t + 9\sin^2t}\right]^3=\left[\sqrt{{\color{red}4} + 5\sin^2t}\right]^3$

I don't think this last expression can be further simplified.

5. Sorry flyingsquirrel may i know why the equation cannot be simplified this way ? Learning along the way

$\displaystyle [\sqrt{4(cos^2t) + 9(sin^2t)} ]^3$

$\displaystyle [2cost + 3sint]^3$

$\displaystyle 8cos^3t + 27sin^3t$

6. Originally Posted by tester85
Sorry flyingsquirrel may i know why the equation cannot be simplified this way ? Learning along the way

$\displaystyle [\sqrt{4(cos^2t) + 9(sin^2t)} ]^3$

$\displaystyle [2cost + 3sint]^3$

Wooo, according to you, it is correct to write something like $\displaystyle 5=\sqrt{25}=\sqrt{9+16}= \sqrt{3^2+4^2}=3+4=7$ ?!

The problem is that, in general, $\displaystyle \sqrt{a^2+b^2}$ does not equal $\displaystyle a+b$ : you can't say that $\displaystyle \sqrt{4 \cos^2t + 9\sin^2t}=2\cos t+3\sin t$ and you can't say that $\displaystyle \sqrt{3^2+4^2}=3+4$.

However, it is correct to write $\displaystyle \left[\sqrt{x}\right]^2=x$ or $\displaystyle \sqrt{x^2}=|x|$... but these two identities can't be used here.

$\displaystyle [2cost + 3sint]^3$

$\displaystyle 8cos^3t + 27sin^3t$

Again, in general, $\displaystyle (a+b)^3$ does not equal $\displaystyle a^3+b^3$. Remember that $\displaystyle (a+b)^3=(a+b)(a+b)(a+b)$. If you expand this, you'll see that $\displaystyle (a+b)^3=a^3+3a^2b+3ab^2+b^3$...