Quick question about simplifying radical expressions

[hackedagainanda]

Homework Statement

Simplify the expression:
$\sqrt{84 \,(y - 2)^ 3\; }$

Relevant Equations

None.

So I simplify the expression and get 2 x 3^2 x 7 = 84 and get $2 \sqrt{21(y−2)^{3/2}}$
I don't follow how the answer is $2(y -2) \sqrt{21 ( y - 2) }$

 

[BvU]

Could it be that the expression has the 3 under the square root ? Like $$\sqrt{84 \,(y - 2)^ 3\; }\ \ ?$$ in which case you can pull out a factor 2 like you did, as well as a factor $\ y-2\$ and then you are left with the given answer !$$2 ( y -2) \sqrt {21(y-2)}$$

PS: Note the preview button

that allows you to check your $\LaTeX$ -- [edit] ah, you found it ...

 

[RPinPA]

That's just noticing that for any number, $x^{3/2} = x^{1 + (1/2)} = x * x^{1/2}$

So $(y - 2)^{3/2}$ can be written as $(y - 2)(y - 2)^{1/2}$.

Oh, I see another issue. $(y - 2)^{3/2}$ shouldn't be under the radical sign. You had $\sqrt{(y - 2)^3}$. That's equal to $(y - 2)^{3/2}$, not $\sqrt{(y - 2)^{3/2}}$.

 

[hackedagainanda]

Sorry for the mess :( The preview button and MathJax are giving me lots of display issues at the moment.

That's just noticing that for any number, $x^{3/2} = x^{1 + (1/2)} = x * x^{1/2}$

So $(y - 2)^{3/2}$ can be written as $(y - 2)(y - 2)^{1/2}$.

That explains it very clearly, thanks! I sometimes forgot that rational expressions can be factored just like integers.

 

[hackedagainanda]

Slightly off-topic but is there some way to become more proficient at this? I feel like I'm starting to lag behind in my College Algebra course. I don't feel like the book has enough practice problems for me.

 

[Mark44]

Slightly off-topic but is there some way to become more proficient at this? I feel like I'm starting to lag behind in my College Algebra course. I don't feel like the book has enough practice problems for me.

Doesn't your textbook have sets of problems at the ends of the sections and chapters? If not, get another textbook to supplement the one in your class. I'm sure there are lots of them listed on Amazon and elsewhere for not much money. And algebra is algebra, so it doesn't matter much which book you get.

 

[FactChecker]

I always recommend Schaum's Outline series. They have a lot of worked examples and exercises with the answers. (There may now be similar things on the internet, but I am not familiar with any.)