Simplifying Expressions With Roots and Exponents

Now we have all these pesky terms with exponents and roots, and they come in all these complicated looking expressions. Square this, cube root that, whatever will we do? Don't worry, it's usually pretty easy to simplify these until they get into a more palatable form, and it's quite satisfying to get them there! Let's learn all the tricks.

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[alonsorobots]

Damn, I feel like following the Mathematics (All of it) playlist, this one took a jump in complexity!

[JaylouRasonabe]

This video took a curve with the my learning progress, but I will never leave this video until I fully grasp the concept.

[commandosolo]

Explanations of the answers:

1. (2x^2 y^4 z^5) ^3 -- As you recall, when cubing elements with exponent we simply multiply. So 2 cubed is 8, x^2 cubed is x^6, y^4 cubed is y^12, and z^5 cubed is z^15.

2. Remember that because the exponent is negative, we get rid of it by inverting the fraction. So now we have (x^2y over xy^2) squared. We multiply the exponents, yielding X^4 Y^2 over X^2 Y^4. That cancels to give us X^2 over Y^2.

3. Let's substitute Y=3, chosen at random. X^ (6+1) over X^(6-1) gives us X^7 over X^5. Cancel out, and we're left with X^2.

4. Good golly Dave, way to crank up the difficulty! This is just mean.

Okay, remember that the "2" in the exponent is also a 2/1. So now we can multiply the exponents. X^1/3 by 2/1 gives X^2/3. The other term, X^2/3s multiplied by 2/1 gives us X^4/3s. So now we have X^2/3 times X^4/3 divided by X^3/2.

Now we go ahead and multiply X^2/3 by X^4/3, which means we add the numerators, giving us X^6/3. (which is the same as X-squared, but in this case we'll convert it to X^4/2, a common denominator.) Now when we divide X^4/2 by X^3/2, we subtract the numerators, yielding X^1/2. Whew!

(At least I think that's how it works. If I've come to the right answers by erroneous methods, by all means show us your method.)

[basketguitar]

On 3x^2√32y^2 - 4y √18x^4, they cut the 32 in half to take its root, and they leave the √2 since you cant do anything with it. They combine 3x • 4y because mutliplication is associative (doesnt matter what order you multiply), so it is valid, and left us with 12x^2y√2
Same thing on the other side.

It took me a few days to understand this, and I don't want other people to face the same issue so hopefully this helps.

[axelmont]

Ok, this video is pretty good, but the jump in difficulty is in big part because some rules of exponents and radicals are assumed in the resolution of the examples, but have not been thoroughly explained in previous videos in the series. I recommend Nerdstudy's playlist on Algebra I where he goes through each rule. That made this video a lot clearer to me.

[CartoonKidOLLY]

I was doing fine up until this point...

[DaveyLers]

Damn, professor Dave aint fucking around this time....

[Mythtongue]

The difficulty jump is wild. Yet, not unwanted. I know with time and repetition i'll be able to simplify with ease!

[brandonampang]

This is the point in 7th grade where I realized math is getting harder

[afrinshahnaz]

Explanation of 2nd and 4th one:
2nd ques-
(xy2/x2y)^-2
= (x2y/xy2)^2 (becoz there is -2 whole power, so to make it positive we have to reverse it, as x-1=1/x)
=x4y2/x2y4 (now cut x2/x4 and y2/y4)
=>x2/y2 final ans.

4th ques:
(X⅓ . X⅔)^2\X³/²
=(X⅓×2 . X⅔×2)/X³/² (solve power individually )
=(X⅔. X⁴/³)/X³/² (now add numerator powers 2/3+4/3)
=X⁶/³ divided by X³/² ( now cut 6/3)
=X2-³/² =X²/¹ --- ³/²
=X4-3/2 => X1/2 (final ans)
Thank you for listening my Explanation 💜

[bashdota]

I'm 26years old, looking forward to completing the playlist to get prepared for engineering education that I'm willing to start soon. This playlist been perfect so far. I'd only wish if there were more comprehension questions in the video or linked to it. Thanks a lot prof!

[Student_ye]

This was a tuff vid, but just a few more replaying this video. I got it now thanks!!

[alexandrugrancea]

Explanation of the third exercise:
So we have x^(2y+1)/x^(2y-1)
The first term can also be written as x^2y*x^1
The second one as : x^2y*x-1
So we have : x^(2y+1)-(2y-1) taking into consideration that when we have minus before a parathesis we multiply the terms by -1 therefore the +2y cancels with -2y and because -1 became +1 we remain with +1+1 => x^2
The last one it s even essier if you dont do the mistake i did and start as you should by calculating the parathesis, by doing so you will have x^2/x^3/2 which equals x^1/2

[MarianaLafoskayl]

Bro it's so complex but u made it easy asf!! I really love your videos and still go on w learning mathematics with you!!

[fzunvpe]

man, i really love your videos! i used to hate math but grew to love the subject because your videos made me realize that math isn't that bad at all. thanks a lot, professor dave!

[FrancescoDiSiena]

in check comprehension, in the second in the first line i got x^2y^-2, but if you change the y^-2 in 1/y^2, and then you do x^2/1 times 1/y^2, the result is the same. in the last, i just wrote x^1/2 like square root of x, so is the same

[goabeaches]

I saw Leonardo Da Vinci's biography and came to know how a person can be jack of all trades and here I came to learn some maths after getting inspired by Da Vinci!
Professor Dave is making my life much easier. Thank u very much ❤️

[jadelerio]

For practice question #4 is my method right? Pls correct me if not.

x^6/3 divided by x^3/2 but we should make it have a common denominator first...

so: 6/3 x 1/2 and 3/2 x 1/3 which will give us 6/6 - 3/6 or simply 1-1/2

so: x^1/2 or square root is the answer.

[datanerden]

One thing to notice if some people didn't get it: the whole expression in parenthesis changes sign if a minus is in front of it.

[Twichery]

Sure, let's break it down.

So, you have something that looks like this: ((x^(1 part) * x^(2 parts))^2)/x^(3 parts).

Let's start with (x^(1 part) * x^(2 parts)). That means x has 1 part plus 2 parts, which equals 3 parts or just x.

Now we square it (meaning we times it by itself) so now it looks like this: (x * x) or x^2.

In the denominator, we have x^(3 parts). That's like x, but times itself again, so we get 1.5 parts (x^1.5).

Finally, we subtract the bottom part from the top part. So, 2 parts (from x^2) - 1.5 parts (from x^1.5) equals 0.5 parts or x^0.5.

So at the end, you get x^0.5. And that's the answer! You can also say "x to the power of half" or "square root of x".