so! first off, it's important to understand the idea that if you have a bunch of things multiplied together under a root:
√abc
you can take the root of EACH of those parts, and multiply them together.
√abc = √a√b√c
so if you understand that, simplifying your problems are pretty straightforward. please let me know if you need the above idea clarified.
so, now looking at your specific problems:
1. √56x^2
first, we see two things multiplied together, x^2 and 56. let's take the root of EACH part.
=√56√x^2
let's first talk about simplifying √56
to do this, we need to do the prime factorization of what's within, or 56. this just means splitting up 56 into a bunch of prime numbers multiplied together. usually this is done with a "tree".
56 can first be split up into 8 and 7, since 8×7 is 56. 7 is prime so we're done there. 8 is NOT prime though, so let's split it up further. 8 can be split up into 4 and 2, and 4 can be split to 2 and 2. I WOULD draw the tree here, but I can't haha. basically:
56= 8×7
56= 4×2×7
56=2×2×2×7
since both 2 and 7 are prime, this is the prime factorization of 56. alright! from this, we can say:
√56 = √2×2×2×7
I'm gonna do something now, please comment if this doesn't make sense. I'm gonna take two of those 2's and put them under a root by themselves.
√2×2×2×7 = √2×2√2×7
remember our rule--if things are multiplied, we can pull them apart. However, why did we do this? well, notice what 2×2 equals. It's 4, right? And what is the square root of 4? 2!
so,
√2×2×2×7 = 2√2×7
to summarize this, if you have a number under a SQUARE root, your gonna split up that number until you get a "perfect square". then, you're gonna simplify that part, and leave the rest under a SQUARE root. that's how you simplify something like this. so,
√56 = 2√2×7 = 2√14
alright! So, first part done. now we gotta do
√x^2
remember our √2×2? it ended up being 2 right, or the number being multiplied under the square root. apply this idea to the variable. we're taking the square root of x^2. so, were looking for a value, when multiplied by itself, gives us x^2. I'm just gonna say it, but if you need more explanation id be happy to go more in depth. the answer here would just be x.
so! finally:
√56x^2 = 2x√14 or B
I would be happy to assist with the rest of your questions, but I believe you have the tools now to do so. let me know if you need more help!
