Factoring /w rational powers: 5x^2(2x-3)^(1/3)(3x+2)^(1/2)+8x(2x-3)^(-2/3)(3x+2)^(3/2

[prepforcalc]

Hi Everyone,

This is my first post on the Free Math Help website. I am new. I registered to find help reviewing algebra and trig for an upcoming Calculus 1 course. Thus, I am on winter break and not enrolled in anything right now. I will be asking questions purely to self educate prior to spring semester which will not begin until the very end of January.

Thanks in advance for your help! I will be helpful if I can.

Below is my first question.

Can you explain how they get from the first step to each subsequent step here:

. . .\(\displaystyle 5x^2\, (2x\, -\, 3)^{1/3}\, (3x\, +\, 2)^{1/2}\, +\, 8x\, (2x\, -\, 3)^{-2/3}\, (3x\, +\, 2)^{3/2}\)

. . . . .\(\displaystyle =\, x\, (2x\, -\, 3)^{-2/3}\, (3x\, +\, 2)^{1/2}\, \left[\, 5x\, (2x\, -\, 3)\, +\, 8\, (3x\, +\, 2)\, \right]\)

. . . . .\(\displaystyle =\, x\, (2x\, -\, 3)^{-2/3}\, (3x\, +\, 2)^{1/2}\, \left[\, 10x^2\, -\, 15x\, +\, 24x\, +\, 16\,\right]\)

. . . . .\(\displaystyle =\, x\, (2x\, -\, 3)^{-2/3}\, (3x\, +\, 2)^{1/2}\, \left(10x^2\, +\, 9x\, +\, 16\right)\)

Edit: Actually, can anyone recommend an excellent review, such as a pdf or youtube for this purpose of preparing for Calculus 1?

 

[mmm4444bot]

Can you explain how they get from the first step to each subsequent step here:

. . .\(\displaystyle 5x^2\, (2x\, -\, 3)^{1/3}\, (3x\, +\, 2)^{1/2}\, +\, 8x\, (2x\, -\, 3)^{-2/3}\, (3x\, +\, 2)^{3/2}\)

. . . . .\(\displaystyle =\, x\, (2x\, -\, 3)^{-2/3}\, (3x\, +\, 2)^{1/2}\, \left[\, 5x\, (2x\, -\, 3)\, +\, 8\, (3x\, +\, 2)\, \right]\)

. . . . .\(\displaystyle =\, x\, (2x\, -\, 3)^{-2/3}\, (3x\, +\, 2)^{1/2}\, \left[\, 10x^2\, -\, 15x\, +\, 24x\, +\, 16\,\right]\)

. . . . .\(\displaystyle =\, x\, (2x\, -\, 3)^{-2/3}\, (3x\, +\, 2)^{1/2}\, \left(10x^2\, +\, 9x\, +\, 16\right)\)

Edit: Actually, can anyone recommend an excellent review, such as a pdf or youtube for this purpose of preparing for Calculus 1?

Did you really intend to imply that you don't understand how to go from:

5x(2x - 3) + 8(3x + 2)

to:

10x^2 - 15x + 24x + 16

to:

10x^2 + 9x + 16

Nothing else is changing, in those last three lines. They used the Distributive Property, to carry out the multiplications, and then they combined like terms. (We call this process "simplifying an expression".)

Going from the first line to the second, they pulled out common factors from both sides of the outer sum. (Note that they pulled out the factors having the smallest exponent, and the result was that the exponents inside the square brackets are both 1 -- which we don't write.)

For review sites, I can't make a recommendation because I don't know your background or what types of sites you'll find useful.

Try googling keywords precalculus algebra and trigonometry review, and check out some links, to see whether you think any of those sites will serve your needs. :cool:

 

[prepforcalc]

Did you really intend to imply that you don't understand how to go from:

5x(2x - 3) + 8(3x + 2)

to:

10x^2 - 15x + 24x + 16

to:

10x^2 + 9x + 16


Nothing else is changing, in those last three lines. They used the Distributive Property, to carry out the multiplications, and then they combined like terms. (We call this process "simplifying an expression".)

Going from the first line to the second, they pulled out common factors from both sides of the outer sum. (Note that they pulled out the factors having the smallest exponent, and the result was that the exponents inside the square brackets are both 1 -- which we don't write.)

For review sites, I can't make a recommendation because I don't know your background or what types of sites you'll find useful.

Try googling keywords precalculus algebra and trigonometry review, and check out some links, to see whether you think any of those sites will serve your needs. :cool:

Okay, now I understand the last 3 lines which seem amazingly simple now that you explain it. Maybe I've been studying too long, but I actually put time and effort into those lines and didn't see the relationship until now. Maybe I am very rusty. Can still use help getting from line 1 to line 2. Thank you!

 

[mmm4444bot]

Can still use help getting from line 1 to line 2.

Do you remember the basic concept of factoring? If so, then check out some lessons, by googling keywords factoring expressions with rational exponents.

If you're not sure how to factor simpler expressions, like 8x^2-4x, then I suggest that you start your factoring review there. Again, use Google, to find sites like this one, covering "basic factoring".

Let us know, if you need help understanding something in the examples you study. :cool:

 

[stapel]

Can you explain how they get from the first step to each subsequent step here:

. . .\(\displaystyle 5x^2\, (2x\, -\, 3)^{1/3}\, (3x\, +\, 2)^{1/2}\, +\, 8x\, (2x\, -\, 3)^{-2/3}\, (3x\, +\, 2)^{3/2}\)

. . . . .\(\displaystyle =\, x\, (2x\, -\, 3)^{-2/3}\, (3x\, +\, 2)^{1/2}\, \left[\, 5x\, (2x\, -\, 3)\, +\, 8\, (3x\, +\, 2)\, \right]\)

. . . . .\(\displaystyle =\, x\, (2x\, -\, 3)^{-2/3}\, (3x\, +\, 2)^{1/2}\, \left[\, 10x^2\, -\, 15x\, +\, 24x\, +\, 16\,\right]\)

. . . . .\(\displaystyle =\, x\, (2x\, -\, 3)^{-2/3}\, (3x\, +\, 2)^{1/2}\, \left(10x^2\, +\, 9x\, +\, 16\right)\)

Edit: Actually, can anyone recommend an excellent review, such as a pdf or youtube for this purpose of preparing for Calculus 1?

While many sites (including the one in the link, provided in a previous response) arguably do a good job of explaining factoring, I'm not familiar with anything that covers this type of factoring. This is something you will see often ("too often"?) in calculus, but it's beyond the complexity of most ("nearly all"?) algebra classes. Pretty much the best "review" is to do a bunch of these. If you've got a list of review exercises that look like the one above, then do all of them. (If you don't have the answers to some or all of the exercises, show your work here, and we'll be glad to check it.)

It may also help to convert things to radical form, at least until you get used to this sort of computation. For instance:

. . .\(\displaystyle 5x^2\, (2x\, -\, 3)^{1/3}\, (3x\, +\, 2)^{1/2}\, +\, 8x\, (2x\, -\, 3)^{-2/3}\, (3x\, +\, 2)^{3/2}\)

. . . . .\(\displaystyle =\, 5x^2\, \sqrt[3]{\strut 2x\, -\, 3\,}\,\sqrt{\strut 3x\, +\, 2\,}\, +\, \dfrac{8x\, \sqrt{\strut (3x\, +\, 2)^3\,}}{\sqrt[3]{\strut (2x\, -\, 3)^2\,}}\)

. . . . .\(\displaystyle =\, \dfrac{5x^2\, \sqrt[3]{\strut 2x\, -\, 3\,}\,\sqrt[3]{\strut (2x\, -\, 3)^2\,}\, \sqrt{\strut 3x\, +\, 2\,}}{\sqrt[3]{\strut (2x\, -\, 3)^2\,}}\, +\, \dfrac{8x\, (3x\, +\, 2)\, \sqrt{\strut 3x\, +\, 2\,}}{\sqrt[3]{\strut (2x\, -\, 3)^2\,}}\)

. . . . .\(\displaystyle =\, \dfrac{5x^2\, \sqrt[3]{\strut (2x\, -\, 3)^3\,}\, \sqrt{\strut 3x\, +\, 2\,}}{\sqrt[3]{\strut (2x\, -\, 3)^2\,}}\, +\, \dfrac{8x\, (3x\, +\, 2)\, \sqrt{\strut 3x\, +\, 2\,}}{\sqrt[3]{\strut (2x\, -\, 3)^2\,}}\)

. . . . .\(\displaystyle =\, \dfrac{5x^2\, (2x\, -\, 3)\, \sqrt{\strut 3x\, +\, 2\,}}{\sqrt[3]{\strut (2x\, -\, 3)^2\,}}\, +\, \dfrac{8x\, (3x\, +\, 2)\, \sqrt{\strut 3x\, +\, 2\,}}{\sqrt[3]{\strut (2x\, -\, 3)^2\,}}\)

. . . . .\(\displaystyle =\, \left(\dfrac{x\, \sqrt{\strut 3x\, +\, 2\,}}{\sqrt[3]{\strut (2x\, -\, 3)^2\,}}\right)\, \left(\dfrac{5x\, (2x\, -\, 3)}{1}\right)\, +\, \left(\dfrac{x\, \sqrt{\strut 3x\, +\, 2\,}}{\sqrt[3]{(2x\,-\, 3)^2\,}}\right)\, \left(\dfrac{8\, (3x\, +\, 2)}{1}\right)\)

Now do you see how the factoring happened?

 

[stapel]

By the way, it took me a looooooong time of switching back and forth between fractional powers and radicals, before I got to where I could "see" how they went from their first line to their second. Don't worry! If you give yourself enough time and enough practice, you'll get there!

 

[Deleted member 4993]

Hi Everyone,

This is my first post on the Free Math Help website. I am new. I registered to find help reviewing algebra and trig for an upcoming Calculus 1 course. Thus, I am on winter break and not enrolled in anything right now. I will be asking questions purely to self educate prior to spring semester which will not begin until the very end of January.

Thanks in advance for your help! I will be helpful if I can.

Below is my first question.

Can you explain how they get from the first step to each subsequent step here:

. . .\(\displaystyle 5x^2\, (2x\, -\, 3)^{1/3}\, (3x\, +\, 2)^{1/2}\, +\, 8x\, (2x\, -\, 3)^{-2/3}\, (3x\, +\, 2)^{3/2}\)

You see that x , (2x-3) and (3x+2) are common in both the addendends

The minimum powers of each of the terms were → (1), (-2/3) and (1/2) - so I want to factor out → x*(2x-3)^(-2/3) * (3x+2)^(1/2)

Give it a try.....

. . . . .\(\displaystyle =\, x\, (2x\, -\, 3)^{-2/3}\, (3x\, +\, 2)^{1/2}\, \left[\, 5x\, (2x\, -\, 3)\, +\, 8\, (3x\, +\, 2)\, \right]\)

. . . . .\(\displaystyle =\, x\, (2x\, -\, 3)^{-2/3}\, (3x\, +\, 2)^{1/2}\, \left[\, 10x^2\, -\, 15x\, +\, 24x\, +\, 16\,\right]\)

. . . . .\(\displaystyle =\, x\, (2x\, -\, 3)^{-2/3}\, (3x\, +\, 2)^{1/2}\, \left(10x^2\, +\, 9x\, +\, 16\right)\)

Edit: Actually, can anyone recommend an excellent review, such as a pdf or youtube for this purpose of preparing for Calculus 1?

.