Show (2/5)(x-1)^(5/2)+(2/3)(x-1)^(3/2) equiv to (2/3)x(x-1)^

[jlmills5]

The problem: Show that these two equations equal using algebraic manipulation

(2/5)(x-1)^(5/2) + (2/3)(x-1)^(3/2)

and

(2/3)x(x-1)^(3/2) - (2/3)(2/5)(x-1)^(5/2)

I've plugged in numbers to see that the two actually equal and been assured by my Calc teacher that they in fact do. I haven't done much algebra in over 5+ years. I've set the equations equal to one another, tried simplifying in many ways. I've tried multiplying both by (3/2) to simplify and factoring out (x-1) with no luck. I'm stuck. No matter what I try I can't show proof they equal! Help!

 

[galactus]

Re: Show equations equal

Since you have claimed to have given it an earnest effort, here goes:

Put equal exponents on their respective sides:

\(\displaystyle \frac{2}{3}(x-1)^{\frac{3}{2}}-\frac{2}{3}x(x-1)^{\frac{3}{2}}=\frac{-4}{15}(x-1)^{\frac{5}{2}}-\frac{2}{5}(x-1)^{\frac{5}{2}}\)

Factor out \(\displaystyle \frac{2}{3}(x-1)^{\frac{3}{2}}\) out of the left side and add the like terms on the right:

\(\displaystyle \frac{2}{3}(x-1)^{\frac{3}{2}}(1-x)=\frac{-2}{3}(x-1)^{\frac{5}{2}}\)

\(\displaystyle \frac{-2}{3}(x-1)^{\frac{3}{2}}(x-1)=\frac{-2}{3}(x-1)^{\frac{5}{2}}\)

Now, you can see it, can't you?. Remember, when you multiply exponents terms you add the exponents.

When you divide them you subtract the exponents.

 

[jlmills5]

Re: Show equations equal

Amazing. It has been entirely too long since I have done basic algebra...it really never occured to me to combine like variable by moving them to the other side and whatnot. Thank you so much. It was really easy once I realized I could move things around.