How do I know if I successfully completely simplified?

[znick46]

Hi,

I'm working on simplifying radicals, and know that you should factor by GCF to completely simplify. I don't know how to check if I simplified by the GCF.

Example: √180

answer: 6√5
GCF: 36

Is the most efficient/effective way to list out all the factors of 180 from 1-10?

Thank You
Nick

 

[pka]

I'm working on simplifying radicals, and know that you should factor by GCF to completely simplify. I don't know how to check if I simplified by the GCF.
Example: √180
answer: 6√5
GCF: 36
Is the most efficient/effective way to list out all the factors of 180 from 1-10?

Yes there is a way. Use recourses freely available. Look at this website http://www.wolframalpha.com/input/?i=factor+180

The \(\displaystyle \sqrt{2^2\cdot 3^2\cdot 5}\) tells us that it simplifies to \(\displaystyle 2\cdot3\sqrt5\).

 

[HallsofIvy]

180 is clearly even so has 2 as a factor: 180= 2(90). 90 is even: 180= 2(2)(45). 45 is divisible by 3: 180= 2(2)(3)(15). 15 is divisible by 3: 180= 2(2)(3)(3)(5). 5 is a prime number so we are done.

Since you asked about factors, I looked at prime number factors in turn: 2, 3, 5, 7, ...

But since your object here is to find the square root for this particular problem, it is sufficient to look at squares of primes: 4, 9, 25, 49, ...
180= 4(45)= 4(9)(5) and since 5 is prime we are done.

 

[znick46]

Thanks Guys

 

[Dale10101]

Notational method for prime factoring

Hi,

I'm working on simplifying radicals, and know that you should factor by GCF to completely simplify. I don't know how to check if I simplified by the GCF.

Example: √180

answer: 6√5
GCF: 36

Is the most efficient/effective way to list out all the factors of 180 from 1-10?

Thank You
Nick

I use a simple notational method of prime factoring integers that goes like this:

\[\begin{array}{l}
180 - > \frac{{180}}{2} - > \frac{{90}}{2} - > \frac{{45}}{3} - > \frac{{15}}{3} - > \frac{5}{5} - > 1\\
{\rm{so }}...{\rm{ }}180 = ({2^2})({3^2})(5)
\end{array}\]

In simple cases like this you can write the string out as fast as your fingers can move.

One starts with the number to be prime factored and divides by the lowest prime number which divides evenly and works rightward, (after exhausting that prime number) by trying the next lowest prime number. I like the method because you can see everything at once. I use an arrow because it shows the flow direction and because "=" would be inappropriate. No doubt others have devised something similar and who knows maybe I saw it elsewhere sometime.