Simplify 192*pi^2 / (128*pi + 32*pi^2).

Step-by-step solution

1. 1

   Do not cancel term by term. A denominator that is a sum must be factored before anything cancels; striking $32\pi^{2}$ against part of the numerator would be invalid. So the first job is to find the greatest common factor of the two denominator terms.

2. 2

   Factor the denominator. $128=32\cdot 4$ and $32\pi^{2}=32\pi\cdot\pi$, so both terms share $32\pi$: $128\pi+32\pi^{2}=32\pi\left(4+\pi\right).$

3. 3

   Factor the numerator to expose the same factor. $192=32\cdot 6$, hence $192\pi^{2}=32\pi\cdot 6\pi.$ Writing it this way makes the shared factor explicit rather than implicit.

4. 4

   Cancel the common factor. $\frac{32\pi\cdot 6\pi}{32\pi\left(4+\pi\right)}=\frac{6\pi}{4+\pi},$ valid because $\pi\neq 0$. Nothing remains to cancel: $6\pi$ and $4+\pi$ share no factor, since $4+\pi$ is not a multiple of $\pi$.

5. 5

   Check numerically. With $\pi=3.14159265$, the original is $\dfrac{192(9.86960440)}{128(3.14159265)+32(9.86960440)}=\dfrac{1894.964045}{402.1238597+315.8273408}=\dfrac{1894.964045}{717.9512005}=2.639405$, and the simplified form gives $\dfrac{6(3.14159265)}{7.14159265}=\dfrac{18.84955592}{7.14159265}=2.639405$. The two agree to six decimals.