Add, sub, multiply, and dividing w/ fractional exponents & radicals

[Dr_Kesha_PHD]

Okay so I'm in Calculus 1 and we are working on derivatives. I understand it all but I have been having some trouble with some basic math skills that I cannot remember from high school and I can't seem to find a good tutorial anywhere online.

I am having problems with multiplying fractional exponents and also adding them. I know that when you have multiply or divide exponents it is always like x^6*x^2=x^6+2=x^8.

But it is throwing me off when the exponents have a fraction or radical because I don't remember these basic skills from high school.

Let me give you an example of a problem that I am struggling with...

Differentiate:
H(u)=(u-sqrt(u))(u+sqrt(u))

I know that this means to find the derivative of this equation. So I use the product rule which is the derivative of the 1st term multiplied by the second term, plus the derivative of the second term multiplied by the first time.
This gives me:

(u-sqrt(u))(1+1/2u^-1/2)+(u+sqrt(u))(1-1/2u^-1/2)

How do I perform -sqrt(u) multiplied by 1/2u^-1/2 ?

Can someone show me where I am going wrong because I am coming up with -1/2.

Please help explain or link me to a video to me how to add, multiply, subtract and diving when I have radicals/fractional exponents like this.

 

[micromass]

How do I perform -sqrt(u) multiplied by 1/2u^-1/2 ?

Can someone show me where I am going wrong because I am coming up with -1/2.

That is correct:

$$-\frac{1}{2}\sqrt{u} u^{-1/2} = -\frac{1}{2} u^{1/2} u^{-1/2} = -\frac{1}{2} u^{1/2 - 1/2} = -\frac{1}{2}u^0 = -\frac{1}{2}$$

 

[Dr_Kesha_PHD]

That is correct:

$$-\frac{1}{2}\sqrt{u} u^{-1/2} = -\frac{1}{2} u^{1/2} u^{-1/2} = -\frac{1}{2} u^{1/2 - 1/2} = -\frac{1}{2}u^0 = -\frac{1}{2}$$

Well then I'm not totally sure where I am going wrong because I keep coming up with the wrong answer to the problem. But thank you very much for showing me that that wasn't where I am making my mistake.

 

[HallsofIvy]

Okay so I'm in Calculus 1 and we are working on derivatives. I understand it all but I have been having some trouble with some basic math skills that I cannot remember from high school and I can't seem to find a good tutorial anywhere online.

I am having problems with multiplying fractional exponents and also adding them. I know that when you have multiply or divide exponents it is always like x^6*x^2=x^6+2=x^8.

But it is throwing me off when the exponents have a fraction or radical because I don't remember these basic skills from high school.

Let me give you an example of a problem that I am struggling with...

Differentiate:
H(u)=(u-sqrt(u))(u+sqrt(u))

Rather than use the "product rule", I would just go ahead and multiply:
$H(u)= u^2- u$

So H'(u)= 2u- 1

I know that this means to find the derivative of this equation. So I use the product rule which is the derivative of the 1st term multiplied by the second term, plus the derivative of the second term multiplied by the first time.
This gives me:

(u-sqrt(u))(1+1/2u^-1/2)+(u+sqrt(u))(1-1/2u^-1/2)

How do I perform -sqrt(u) multiplied by 1/2u^-1/2 ?

Can someone show me where I am going wrong because I am coming up with -1/2.

Please help explain or link me to a video to me how to add, multiply, subtract and diving when I have radicals/fractional exponents like this.[/QUOTE]

 

[CellCoree]

I can understand your confusion. Working with fractional exponents and radicals can be challenging, especially if you haven't practiced them in a while. But don't worry, with some practice and understanding of the rules, you'll be able to solve these problems easily.

First, let's review the basic rules of exponents. When multiplying exponents with the same base, we add the exponents. For example, x^2 * x^3 = x^(2+3) = x^5. When dividing exponents with the same base, we subtract the exponents. For example, x^5 / x^2 = x^(5-2) = x^3.

Now, let's apply these rules to fractional exponents. When multiplying fractional exponents with the same base, we add the numerators and denominators separately. For example, x^(3/5) * x^(2/5) = x^(3+2)/5 = x^5/5 = x^1. Similarly, when dividing fractional exponents with the same base, we subtract the numerators and denominators separately. For example, x^(3/5) / x^(2/5) = x^(3-2)/5 = x^1/5.

In your example, the term -sqrt(u) can be written as -(u^(1/2)). So when multiplying it by 1/2u^(-1/2), we use the rule for multiplying fractional exponents and get -(u^(1/2+(-1/2))) = -(u^0) = -1.

I hope this explanation helps you understand how to perform operations with fractional exponents and radicals. For further practice, I recommend checking out online tutorials or practice problems. Khan Academy and MathIsFun are great resources for learning math concepts. Best of luck with your calculus studies!