1. ## Simplifying exponential expressions

Simplifying exponential expressions, can you please confirm my solutions are correct?

A.

(3x^4y^5z^7)^5 / (-3x^3yz^4)^7

= -1^5-7x^20-21y^18z^7 = -1^-2x^-1y^18z^7

Solution I arrived at:

y^18z^7/ x

B.

[x^(a+b)]^(a-b) / [x^(a-2b)]^(a+2b)

Solution:
= 1^(3b^2)

Sincerely,

Raymond MacNeil

2. Just to be clear, the original expressions are:

A.

$\frac{(3x^{4}y^{5}z^{7})^{5}}{(-3x^{3}yz^{4})^{7}},$ and

B.

$\frac{[x^{a+b}]^{a-b}}{[x^{a-2b}]^{a+2b}}.$

Is that right?

3. That's correct sir. I tried importing images from Microsoft Equation editor but they didn't seem to want to upload.

4. Fantastic. The most important two concepts you need to solve this problem are the following:

1. $(a^{b})^{c}=a^{bc},$ and

2. $\frac{1}{a^{b}}=a^{-b}.$

I would probably apply # 1 first. What does that give you?

5. for the First one I have now arrived at the solution: -y^18z^7/9x

Is that correct? I'll work on the second one again.

6. I would agree with your answer, since I know what you meant to write. However, you should technically write -(y^18)(z^7)/(9x) to be completely unambiguous. Don't write so that people can understand you: write so they can't misunderstand you!

7. Your solution to the second is incorrect.

\displaystyle \begin{align*}\frac{\left(x^{a + b}\right)^{a - b}}{\left(x^{a-2b}\right)^{a+2b}} &= \frac{x^{(a + b)(a - b)}}{x^{(a + 2b)(a - 2b)}}\\ &= \frac{x^{a^2 - b^2}}{x^{a^2 - 4b^2}} \end{align*}.

Can you go from here?

8. I guess I get baffled here because I am not entirely sure what to do when you have positive and negative exponents. Do you turn it into a complex fraction? I substituted numbers for the the unknowns and it seems to work out that the correct solution would be x^(3b^2). I acknowledge previously I had a "1" instead of maintaining the variable "x". Though perhaps I am still doing something wrong. Do you mind showing me the entire process because I have never encountered this type of expression before and the course has not supplied any examples demonstrating how to carry this out either.

Sincerely,

Raymond.

9. x^(3b) is correct.

10. Originally Posted by Prove It
x^(3b) is correct.
So the answer is not x^(3b^2)?

11. ProveIt had a typo, or maybe a thought-o in post # 9. The correct answer is

$x^{3b^{2}}.$

12. Thanks, yeah I edited right after once I realized the ambiguity. Thanks for confirming.

13. You're welcome!