Math Help: factoring, dividing poly's, simplifying, etc.

[Paris]

1) Factor out the greatest common factor:

. . .3x^6 - 24x^5 + 18x^4

What I got was is:

. . .3x^4(x^2 - 8x + 6)

Did I do that right?

2) Divide and write the answer without exponents:

. . .(4 * 10^5) / (2 * 10^(-2))

I have no clue what to do.

3) Use the exponent rule to simplify the expression. Assume the variables represent nonzero real numbers.

. . .[(m^8 n)^(-2)] / [m^(-4) n^7]

I don't know what to do for this one.

4) Perform the indicated operation, and simplify your answer:

. . .[(x^2 - 3x - 18) / (x^2 - 6x + 9)] / [(x - 6) / (x - 3)]

Again...I don't know what to do.

 

[tkhunny]

1) Let's see

3/3 = 1
24/3 = 8
18/3 = 6
6-4 = 2
5-4 = 1
4-4=0

Perfect. This leads me to believe that you have more clue than you are suggesting on the others.

Helpful Pieces:

4/2 = 2
10^5 / 10^(-2) = 10^(5-(-2)) = 10^(5+2) = 10^7 = 10*10*10*10*10*10*10

 

[stapel]

We can't teach lessons within this environment, so it would help if you could narrow down just where you're having difficulty. Since this content must have been covered in class and in your textbook, you must have some idea what to do. What have you tried? Where are you stuck?

1) This factoring is correct.

2) What is 4 / 2? (This is just grade-school division.) What is 10⁵ / 10^-2? (This is just a simple application of the exponent rules you learned.)

3) "What to do" is follow the instructions by applying the exponent rules you've learned, namely:

. . . . .x^mxⁿ = x^m+n

. . . . .x^m / xⁿ = x^m-n

. . . . .(x^m)ⁿ = x^mn

4) To handle the division by a fraction, do what you learned back in grammar school: flip the second fraction, and convert the division to multiplication. To simplify the multiplication, first cancel off any common factors. To do this, factor the quadratics by whatever method they taught you in class.

If you get stuck, please reply showing what you have tried and how far you have gotten. Thank you.

Eliz.

 

[soroban]

Hello, Paris!

1) Factor out the greatest common factor: .\(\displaystyle 3x^6 \,-\,24x^5\,+\,18x^4\)

What I got was: .\(\displaystyle 3x^4(x^2\,-\,8x\,+\,6)\;\;\)Did I do that right? . . . Yes!

2) Divide and write the answer without exponents: .\(\displaystyle \L\frac{4\cdot10^5}{2\cdot10^{-2}}\)

We have: \(\displaystyle \L\:\frac{4}{2}\,\cdot\,\frac{10^5}{10^{-2}}\)\(\displaystyle \;= \;2\,\cdot\,10^7\;=\;20,000,000\)

3) Use the exponent rules to simplify the expression: .\(\displaystyle \L\frac{(m^8n)^{-2}}{m^{-4}n^7}\)

We have: \(\displaystyle \L\:\frac{(m^8)^{-2}(n)^{-2}}{m^4n^7} \:=\:\frac{m^{-16}n^{-2}}{m^4n^7} \:=\:\frac{1}{m^{20}n^9}\)

4) Perform the indicated operation, and simplify your answer:
. . \(\displaystyle \L\frac{x^2\,-\,3x\,-\,18}{x^2\,-\,6x\,+\,9}\,\div\,\frac{x\,-\,6}{x\,-\,3}\)

To divide by a fraction, invert the second fraction and multiply.

We have: \(\displaystyle \L\:\frac{x^2\,-\,3x\,-\,18}{x^2\,-\,6x\,+\,9}\,\cdot\,\frac{x\,-\,3}{x\,-\,6}\)

Factor: \(\displaystyle \L\:\frac{(x\,+\,3)(x\,-\,6)}{(x\,-\,3)(x\,-\,3)}\,\cdot\,\frac{x\,-\,3}{x\,-\,6}\)

Reduce: \(\displaystyle \L\:\frac{(x\,+\,3)(\sout{x\,-\,6})}{(x\,-\,3)(\sout{x\,-\,3})}\,\cdot\,\frac{\sout{x\,-\,3}}{\sout{x\,-\,6}} \;=\;\frac{x\,+\,3}{x\,-\,3}\)

 

[Paris]

Thank you everyone.