Master Fraction Math Problems with Expert Help

Simplifying this algebraic fraction
I don't understand how to simplify this fraction:
$\frac{12 a b - 12 b}{8 a c - 24 c}$
This is my idea to what should be done, but I think it is totally wrong:
$\frac{12 a b - 12 b}{8 a c - 24 c} = \frac{12 \cdot a \cdot b - 12 \cdot b}{8 \cdot a \cdot c - 24 \cdot c} = \frac{12 b^{2} - 12 b}{8 c - 24 c} = \frac{b}{8 c - 24 c} = ?$
Edit: I want to simplify the fraction.

Jazmine Sweeney 2022-04-25

Simplifying fractions - Ending up with wrong sign
I've been trying to simplify this
$1 - \frac{1}{n + 2} + \frac{1}{(n + 2) (n + 3)}$
to get it to that
$1 - \frac{(n + 3) - 1}{(n + 2) (n + 3)}$
but I always end up with this
$1 - \frac{(n + 3) + 1}{(n + 2) (n + 3)}$
Any ideas of where I'm going wrong? Wolfram Alpha gets it to correct form but it doesn't show me the steps (even in pro version)
Thanks

Trace Mcintyre 2022-04-25

Simplifying compound fraction: $\frac{3}{\frac{\sqrt{5}}{5}}$
I'm trying to simplify the following: $\frac{3}{\frac{\sqrt{5}}{5}}$
I know it is a very simple question but I am stuck. I followed through some instructions on Wolfram which suggests that I multiply the numerator by the reciprocal of the denominator.
The problem is I interpreted that as:
$\frac{3}{\frac{\sqrt{5}}{5}} \times \frac{5}{\sqrt{5}},$
Which I believe is:
$\frac{15}{\frac{5}{5}} = \frac{15}{1} .$
What am I doing wrong?

zamenjenot7k 2022-04-25

Simplifying fractions
How to transform
$\frac{{(1 + \frac{\sqrt{1 - 4 a b}}{2} a)}^{n} - {(1 - \frac{\sqrt{1 - 4 a b}}{2} a)}^{n}}{{(1 + \frac{\sqrt{1 - 4 a b}}{2} a)}^{m} - {(1 - \frac{\sqrt{1 - 4 a b}}{2} a)}^{m}}$
into
$\frac{{(\frac{b}{a})}^{n} - 1}{{(\frac{b}{a})}^{m} - 1}$
Thanks.
Added: As per comments, there is an additional assumption that $a, b > 0, b = 1 - a,$ and $b > a .$

livinglife100s8x 2022-04-25

Simplify this expression
$w {(\frac{Q}{1 + \frac{w}{r}})}^{2} + r {(\frac{Q}{1 + \frac{r}{w}})}^{2}$

Malachi Novak 2022-04-24

Is it possible to rationalize a denominator containing two cube roots?
The fraction in question is
$- \frac{12}{\sqrt[3]{12 \sqrt{849} + 108} - \sqrt[3]{12 \sqrt{849} - 108}}$
And was reached in calculating the solution to $x^{4} - x - 1 = 0$ . I've tried all the standard methods, including $(a + b) (a - b) = a^{2} - b^{2}$ , but that doesn't work for cube roots, because once you have the square of one the two middle terms will not cancel each other out.

bacfrancaiso0j 2022-04-23

Negative fractions - what's the difference?
What's the difference between the following fractions:
$\frac{- 4}{- 5}$
$\frac{4}{- 5}$
$\frac{- 4}{5}$
$- \frac{4}{5}$

pacificak8m 2022-04-23

Is it possible to have a fraction wherein the numerator and denominator are also fractions?
For example:
$\frac{\frac{3}{5}}{\frac{7}{8}}$
I was wondering if such a situation had a name, wherein both the numerator and the denominator of a fraction consist of fractions themselves. I was also wondering if this was something common, or at least in some way useful.

Jaidyn Cook 2022-04-23

Interesting pattern in the decimal expansion of $\frac{1}{243}$
There appears to be an interesting pattern in the decimal expansion of
$\frac{1}{243}$
$\frac{1}{243} = 0 . \overset{―}{004115226337448559670781893}$
I was wondering if anyone could clarify how this comes about?

eszkortwxp 2022-04-23

Integers and fractions
How would I write this as an integer or a fraction in lowest terms?
$(1 - \frac{1}{2}) (1 + \frac{1}{2}) (1 - \frac{1}{3}) (1 + \frac{1}{3}) (1 - \frac{1}{4}) (1 + \frac{1}{4}) \dots . . (1 - \frac{1}{99}) (1 + \frac{1}{99})$
I really need to understand where to start and the process if anyone can help me.

misangela4gi 2022-04-22

The intial fraction is:
$\frac{3 x^{3} - x + 4}{x^{3} + 2 x^{2} + 6 x}$
How should to make the decomposition for the denominator?
Thanks!

Andrea Potter 2022-04-21

Solve $\frac{x - a}{b} - \frac{x + c}{d} = 0$ for x
I need to solve
$\frac{x - a}{b} - \frac{x + c}{d} = 0$
for x.
The answer is:
$x = \frac{a d + b c}{d - b}$
But i can't figure out how to get there, I think i have to start by making the fractions have the same denominator but after that I'm stuck.
Thanks in advance

hestyllvql 2022-04-21

Solve algebraically:
$lim_{x \to 3} \frac{3 - x}{5 - \sqrt{x^{2} + 16}}$

Gretchen Barker 2022-04-15

manipulation of subtraction
I am trying to solve an induction problem and got stuck at this part.
$1 - \frac{n + 2}{(n + 2)!} + \frac{n + 1}{(n + 2)!} = 1 - \frac{(n + 2) - (n + 1)}{(n + 2)!}$
Shouldn't it be
$1 - \frac{n + 2}{(n + 2)!} + \frac{n + 1}{(n + 2)!} = 1 - \frac{(n + 2) + (n + 1)}{(n + 2)!}$
How do you get the left expression to the right expression?

Efrain Watkins 2022-04-15

How do I use the partial fractions technique in this case?
$\frac{(x - 1)}{(x^{2} - x + 1) (x + 1)}$

Brielle James 2022-04-14

Is this a weighted average/percentage problem?
Let's say a Marketing company has a total turnover of 10000 $
There are 3 salesmen A,B,C with the following turnovers
$A = 2000 $$
$B = 3000 $$
$C = 5000 $$
Now, If the company wants to increase total turnover by 10 %, Should this be split as 10% for all salesmen OR Can 10% be split across A,B,C differently using weighted averages/percentages?

arbixerwoxottdrp1l 2022-04-12

Why Does $f (x) = x \sqrt{x + 3}$ Only Have One Critical Point?
I am trying to find the critical points of the function $f (x) = x \sqrt{x + 3}$ , then by using the First Derivative Test, determine which ones are a local maximum, local minimum, or neither.
Using the product rule, we get $\frac{3 (x + 2)}{2 \sqrt{x + 3}}$ . So $x = - 2$ is a critical point. But isn't $x = - 3$ also a critical point because if $x = - 3$ then the derivative of $f (x)$ wouldn't exist.
To give an example of why I am thinking this, look at $g (x) = x^{2 / 3} (6 - x)^{1 / 3}$ .
The derivative of $g (x)$ is $\frac{4 - x}{x^{1 / 3} (6 - x)^{2 / 3}}$ , where $x$ cannot equal $0$ or $6$ .
The critical points for $g (x)$ are $0$ , $6$ , and $4$ .
So why aren't $- 2$ and $- 3$ the critical points for $f (x)$ ?

encamineu2cki 2022-04-12

Prove that for every natural number n, fraction $\frac{21 n + 4}{14 n + 3}$ is irreducible. I deduced that if we can prove that numerator and denominator have 1 as their GCD, we can get the result, but I cannot get it from thereon.

Tristan Meyers 2022-04-12

Is the field of rational numbers $Q$ isomorphic to the fraction field $Frac (Q [x])$ ?
Both are fields, I can't disprove it by some algebraic properties that hold for one but not for another.
I know that $Frac (Z [x]) ≅ Frac (Q [x])$ so I have tried to find out whether $Frac (Z [x]) ≅ Q$ , but still have no ideas;
I have also tried to define $ϕ : Q \to Frac (Q [x])$ by
$ϕ (p / q) = (x^{p} - 1) / (x^{q} - 1)$
but it is not a ring homomorphism.
I guess that if $ϕ$ is a ring homomorphism, then any $p / q$ should be mapped to some form of fraction of polynomials $p (x) / q (x)$ such that the product and sum of two polynomials of those form should have the same form, but I don't know how to do.

Annabel Sullivan 2022-04-12

How do you solve $x^{\log x} = 100 x$ ?
Can you please thoroughly explain the left side of the equation.
Please explain very clearly because I have only been learning logarithms for about a week.

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