Reviewing Algebra for CAT6 Test: 3 Challenging Problems

[Imparcticle]

I am currently reviewing my whole algebra textbook for a CAT6 test on tuesday. Apparently, it is very difficult, so I'm taking every minute to review.
Anywho, I ran into 3 problems that are currently driving me crazy.

One concerns roots.
Here is the problem the book presents:
(6x²x)x^1/2

My answer was (36x³)^1/2. But the book says the answer is (6x)^5/2

BTW, how do you write root signs using the latex program?

another root problem:

y³/y^1/4

My answer was x^3/4. The book says x^11/4 is the answer.

Now, here is a factoring problem:
-4(x+16)⁴ + 9(x+16)² + x + 16

On this problem, the part that confuses me is the answer:
[-4(x+16)³ + 9x + 145) (x+16)

I arrived at this answer on my own, except I went one step further:
when I got to [-4(x+16)³ + 9x + 145) (x+16) , I went ahead and rewrote (x+16)³ as (x+16)(x+16)² so that I could distribute the -4. I then arrived at (x+16)<sup>3 (5x+81). Apparently, my final step was incorrect. I would like to know why.

Also, I did most of these operations in my head (as it is easier for me to do so), so if you would like me to explain anything that I did, I should point out that I may sound unmathematical in a sense.

Thank you for your time.</sup>

 

[Muzza]

The book is wrong on the first problem, and so are you. ;) Remember the rule that says $$a^xa^y = a^{x + y}$$. This rule works even when x or y are fractions.

$$(6x^2x)x^{1/2} = (6x^3)x^{1/2} = 6x^3x^{1/2} = 6x^{3 + 1/2} = 6x^{7/2}$$

For the second problem, remember $$\frac{a^x}{a^y} = a^{x - y}$$, even when x or y are fractions.

$$\frac{y^3}{y^{1/4}} = y^{3 - 1/4} = y^{11/4}$$

 

[matt grime]

And the last part on factorization. It's just wrong. You cannot pull a factor of x+16 out of the expression again after you've simplified it once. What do you mean to do in "distributing the -4"? You have done something that is against the laws of arithmetic and algebra, that's all.

 

[Imparcticle]

Muzza: Thank you for your kind explanation. It seems to me that from now on, it is far more efficient to calculate roots exponentially; without using a radical.

Mattgrime: When I said "distribute the -4", I meant applying the distributive property.

But I see the mistake now. They weren't like terms? (I am referring to the factoring problem)