# Simplifying Fractions - Am I right?

• Aug 28th 2006, 12:13 PM
tesla1410
Simplifying Fractions - Am I right?
Instruction: Write a single fraction with one numerator and one denominator.

Problem#1:

1/x + 1/y
I think the answer should be x+y/xy, yes?

Problem #2:

(5a^-1 + 2b^-1) / (3a^-2 - 2b^-2)
I think the answer should be (3a-b) / 5, yes?

Problem #3:
[9-(1/y^2)] / [9+ (6/y) + (1/y^2)]
I think the answer should be 1 / (6/y), yes?

Problem #4 says to factor completely:
18y^3 + 48y^2 + 42y
I think the final answer should be 6y(3y^2 + 8y + 7), yes?

Thank you!
• Aug 28th 2006, 12:20 PM
dan
Quote:

Originally Posted by tesla1410
Instruction: Write a single fraction with one numerator and one denominator.

Problem#1:

1/x + 1/y
I think the answer should be 1/x+y, yes?

$\displaystyle a/b + c/d =[ (ad)+(bc)]/ bd$
• Aug 28th 2006, 12:40 PM
topsquark
Quote:

Originally Posted by tesla1410
Instruction: Write a single fraction with one numerator and one denominator.
(5a^-1 + 2b^-1) / (3a^-2 - 2b^-2)
I think the answer should be (3a-b) / 5, yes?

$\displaystyle \frac{5a^{-1}+2b^{-1}}{3a^{-2}-2b^{-2}} = \frac{\frac{5}{a}+\frac{2}{b}}{\frac{3}{a^2}-\frac{2}{b^2}}$

$\displaystyle = \frac{\frac{5}{a}+\frac{2}{b}}{\frac{3}{a^2}-\frac{2}{b^2}} \cdot \frac{a^2b^2}{a^2b^2} = \frac{ \left ( \frac{5}{a}+\frac{2}{b} \right ) a^2b^2}{ \left ( \frac{3}{a^2}-\frac{2}{b^2} \right ) a^2b^2}$

$\displaystyle = \frac{5ab^2 + 2a^2b}{3b^2 - 2a^2}$

$\displaystyle = \frac{ab(5b + 2a)}{3b^2 - 2a^2}$.

I didn't look at 3, since it's the same method as above, but 1 and 4 look good now.

-Dan
• Aug 28th 2006, 12:49 PM
dan
Quote:

Originally Posted by tesla1410
Problem #4 says to factor completely:
18y^3 + 48y^2 + 42y
I think the final answer should be 6y(3y^2 + 8y + 7), yes?

this looks good
i dont have time now for the others ...maybe some one else can help
dan
• Aug 28th 2006, 01:26 PM
ThePerfectHacker
Quote:

Originally Posted by tesla1410
Problem #3:
[9-(1/y^2)] / [9+ (6/y) + (1/y^2)]
I think the answer should be 1 / (6/y), yes?

$\displaystyle \frac{9-\frac{1}{y^2}}{9+\frac{6}{y}+\frac{1}{y^2}}$
Multiply demoninator and numerator by $\displaystyle y^2$,
$\displaystyle \frac{9y^2-1}{9y^2+6y+1}=\frac{(3y-1)(3y+1)}{(3y+1)(3y+1)}$
Thus,
$\displaystyle \frac{3y-1}{3y+1}$
• Aug 28th 2006, 03:12 PM
tesla1410
Simplifying Fraction - Am I Right? Part 2
You guys are great for helping.

Let me see if I'm getting the hang of it now.

Is this right?

[(x+1)/x] / [(x-3)/4]

To get rid of the fractions, I'll multiply the denominator and numerator by 4x.

The result is 4x + 4 / x^2 - 3x, which can be reduced to

4(x+1)/x(x-3)

Am I right this time?
• Aug 28th 2006, 04:23 PM
ThePerfectHacker
Quote:

Originally Posted by tesla1410
Am I right this time?

Yes
• Aug 28th 2006, 05:25 PM
dan
Quote:

Originally Posted by tesla1410
You guys are great for helping.

Let me see if I'm getting the hang of it now.

Is this right?

[(x+1)/x] / [(x-3)/4]

To get rid of the fractions, I'll multiply the denominator and numerator by 4x.

The result is 4x + 4 / x^2 - 3x, which can be reduced to

4(x+1)/x(x-3)

Am I right this time?

nice work...i think you got it :)
• Aug 28th 2006, 05:56 PM
tesla1410
Simplify Fractions - Am I Right? Part 3
I'm on a roll. I am fairly certain I have the first 2 problems correct, but the last 2 are trickier.

1. [5-(1/a)] / (a+b)

Final answer is (5a-1) / [a(a+b)]

2. [(2x +1) / (x^2 - 25)] / [(4x^2 -1) / (x-5)]

Final answer is (2x + 1) / [(4x^2 -1)(x+5)]

3. {[1/(x+h)] - (1/x)} / h

For this one, I multiplied the denominator & numerator by x(x+h) and eventually ended with a final answer of [x-(x+h)] / [xh(x + h)], which I don't think can be further simplified.

4. (x-y) / [(the square root of x) + (the square root of y)]

I don't think this can be simplified any further. (?)

Thank you for helping me - I am learning a lot.
• Aug 28th 2006, 06:05 PM
Quick
Quote:

Originally Posted by tesla1410
4. $\displaystyle \frac{x-y}{\sqrt x+\sqrt y}$

I don't think this can be simplified any further. (?)

There is a rule in math that says "A fraction is not completely simplified until you rationalize the denominator"
• Aug 28th 2006, 06:07 PM
ThePerfectHacker
Quote:

Originally Posted by tesla1410

3. {[1/(x+h)] - (1/x)} / h

For this one, I multiplied the denominator & numerator by x(x+h) and eventually ended with a final answer of [x-(x+h)] / [xh(x + h)], which I don't think can be further simplified.

The first two did not dim my eye. Perhaps they are correct.
This one did.
It can be simplified.
$\displaystyle [x-(x+h)]=h$

Quote:

(x-y) / [(the square root of x) + (the square root of y)]
It can.

$\displaystyle x-y=(\sqrt{x})^2-(\sqrt{y})^2$
Difference of two squares.
$\displaystyle \frac{(\sqrt{x}-\sqrt{y})(\sqrt{x}+\sqrt{y})}{\sqrt{x}+\sqrt{y}}=$$\displaystyle \sqrt{x}-\sqrt{y}$
• Aug 28th 2006, 06:17 PM
tesla1410
Simplifying Fractions - Part 3
Quote:

Originally Posted by Quick
There is a rule in math that says "A fraction is not completely simplified until you rationalize the denominator"

OK... so let me give this a try.

The original problem is: (x-y) / (square root of x) + (square root of y)

If I rationalize the denominator, I'd be multiplying it and the numerator by the (square root of x) + (square root of y)......right?

So, I end up with:

(x-y)[(square root of x) + (square root of y)] / (x+y)

And this can't be simplified further.

Am I right??
• Aug 28th 2006, 06:22 PM
Quick
Quote:

Originally Posted by tesla1410
OK... so let me give this a try.

The original problem is: (x-y) / (square root of x) + (square root of y)

If I rationalize the denominator, I'd be multiplying it and the numerator by the (square root of x) + (square root of y)......right?

So, I end up with:

(x-y)[(square root of x) + (square root of y)] / (x+y)

And this can't be simplified further.

Am I right??

Hacker solved this, but I would like to pount out that your calculations are wrong anyway...
• Aug 28th 2006, 06:49 PM
ThePerfectHacker
Quote:

Originally Posted by tesla1410
OK... so let me give this a try.

The original problem is: (x-y) / (square root of x) + (square root of y)

If I rationalize the denominator, I'd be multiplying it and the numerator by the (square root of x) + (square root of y)......right?

So, I end up with:

(x-y)[(square root of x) + (square root of y)] / (x+y)

And this can't be simplified further.

Am I right??

If you choose to ignore my method and use rationalization instead:
$\displaystyle \frac{x-y}{\sqrt{x}+\sqrt{y}}\cdot \frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}-\sqrt{y}}$
The denominator is,
$\displaystyle (\sqrt{x}+\sqrt{y})\cdot (\sqrt{x}-\sqrt{y})=x-y$
Thus,
$\displaystyle \frac{(x-y)(\sqrt{x}+\sqrt{y})}{x-y}$