Algebra (all content)
- Simplifying square-root expressions: no variables
- Simplifying square roots of fractions
- Simplifying rational exponent expressions: mixed exponents and radicals
- Simplifying square-root expressions: no variables (advanced)
- Intro to rationalizing the denominator
- Worked example: rationalizing the denominator
- Simplifying radical expressions (addition)
- Simplifying radical expressions (subtraction)
- Simplifying radical expressions: two variables
- Simplifying radical expressions: three variables
- Simplifying hairy expression with fractional exponents
Simplifying radical expressions: two variables
A worked example of simplifying elaborate expressions that contain radicals with two variables. In this example, we simplify √(60x²y)/√(48x). Created by Sal Khan and Monterey Institute for Technology and Education.
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- "why did you convert 5/4 to 1/4*5?" Adding to this question above what videos if any cover the reasons for doing this..I wish to understand it better.
- Because it makes it easier to simplify, because some people get confused with fractions.(4 votes)
- Isn't the xy also divided by 4, like the five is?
- They are all the same term. If they weren't, and the numerator was something like 5+xy you wouldn't be able to, or at least not as cleanly. If that was the case it would be 5/4+xy/4 as you indicated... but since it is all one term: 5xy, its just fine. Heres the reasoning:
5xy/4 is the same as 5xy divided by 4. Division by 4 is the same as multiplying by 1/4. When you do 5xy*1/4 you get 5/4xy. Incase that sounds a bit strange, just think what happens when you multiple 2xy by 4. You would get 8xy, works the same way for fractions!(4 votes)
- I have 1875x^9-48x^5, and i need to factor it. How would I do that? I am soo confused!(3 votes)
- First factor out the GCD:
3x^5 ( 625x^4 - 16)
The binomial is a difference of squares:
3x^5 (25x^2+4)(25x^2 - 4)
The second binomial is also a difference of squares:
Final Answer.(1 vote)
- Wouldn't x be on the outside of the radical sign because x squared is just x? I'm confused. Can someone please explain how the x's cancel kurt/divide. I re-did the question and that was the only part I had trouble with.(1 vote)
- Sal divided the x^2 in the numerator by the x in the denominator, which leaves an x in the numerator because x^2/x is x. So x^2 isn't there anymore.(3 votes)
- is sqrt(a^2 + b^2) the same as sqrt(a^2) + sqrt(b^2)?(1 vote)
- √(a²+b²) = √a² + √b²
Let's see if this works when a=3 and b= 4
√(3² + 4²) must equal √3² + √4²
√9+16 must equal 3+4
√25 must equal 7
But √25 = 5
So √(a²+b²) does not equal √a²+√b²(2 votes)
- Can't you make 5/4 into 1.25, so you just take out the fraction and turn it into a decimal(1 vote)
- You could, but it makes it harder. It is easier to simplify sqrt(5/4) = sqrt(5)/2.(2 votes)
- whats the difference between principal root and square root?(1 vote)
- Principal root is the default and is the non-negative root.
sqrt(9) = 3, not -3
If the problem wants you to use the negative root, then there will be a minus in front of the radical.
- sqrt(9) = -3
Hope this helps.
FYI - Search for intro to square roots. I believe Sal defines the principal root in that vidoe.(2 votes)
- How would you simplify 3x (sqrt(2y))(1 vote)
- Did anyone try to solve this a different way and get an X on the numerator and a radical X on the denominator?(1 vote)
- Would square root 5xy/2 be an equivalent expression?(1 vote)
- Yes, if you multiply 1/2 * sqrt(5xy), you get sqrt(5xy)/2(1 vote)
We're asked to divide and simplify. And we have one radical expression over another radical expression. The key to simplify this is to realize if I have the principal root of x over the principal root of y, this is the same thing as the principal root of x over y. And it really just comes out of the exponent properties. If I have two things that I take to some power-- and taking the principal root is the same thing as taking it to the 1/2 power-- if I'm raising each of them to some power and then dividing, that's the same thing as dividing first and then raising them to that power. So let's apply that over here. This expression over here is going to be the same thing as the principal root-- it's hard to write a radical sign that big-- the principal root of 60x squared y over 48x. And then we can first look at the coefficients of each of these expressions and try to simplify that. Both the numerator and the denominator is divisible by 12. 60 divided by 12 is 5. 48 divided by 12 is 4. Both the numerator and the denominator are divisible by x. x squared divided by x is just x. x divided by x is 1. Anything we divide the numerator by, we have to divide the denominator by. And that's all we have left. So if we wanted to simplify this, this is equal to the-- make a radical sign-- and then we have 5/4. And actually, we can write it in a slightly different way, but I'll write it this way-- 5/4. And we have nothing left in the denominator other than that 4. And in the numerator, we have an x and we have a y. And now we could leave it just like that, but we might want to take more things out of the radical sign. And so one possibility that you can do is you could say that this is really the same thing as-- this is equal to 1/4 times 5xy, all of that under the radical sign. And this is the same thing as the square root of or the principal root of 1/4 times the principal root of 5xy. And the square root of 1/4, if you think about it, that's just 1/2 times 1/2. Or another way you could think about it is that this right here is the same thing as-- so you could just say, hey, this is 1/2. 1/2 times 1/2 is 1/4. Or if you don't realize it's 1/2, you say, hey, this is the same thing as the square root of 1 over the square root of 4, and the square root of 1 is 1 and the principal root of 4 is 2, so you get 1/2 once again. And so if you simplify this right here to 1/2, then the whole thing can simplify to 1/2 times the principal root-- I'll just write it all in orange-- times the principal root of 5xy. And there's nothing else that you can really take out of the radical sign here. Nothing else here is a perfect square.