Master Fraction Math Problems with Expert Help

How could I solve this chain rule problem?
To find: $\frac{d}{d t} [f (c (t))] |_{t = 0}$
Where $c (t)$ is such that $\frac{d}{d t} c (t) = F (c (t))$ , with $F = \nabla f$ , $c (0) = (- \frac{π}{2}, \frac{π}{2})$ and $f (x, y) = \sin (x + y) x y$
I think I would need to proceed using the chain rule but I am currently not sure. How would I go about solving this problem?
Thanks in advance

Michelle Mendoza 2022-07-16

Very Basic Math question?
How can I prove that
$\frac{r}{(1 - x)^{2}} + \frac{r x}{x (1 - x)} = \frac{r}{x (1 - x)^{2}}$
I have tried to prove that , but I could not , can someone help me please ?
Thanks

Roselyn Daniel 2022-07-16

Fraction Decomposition
I have the following problem:
Suppose $x + y + z = 0$ . Show that
$\frac{x^{5} + y^{5} + z^{5}}{5} = \frac{x^{3} + y^{3} + z^{3}}{3} \times \frac{x^{2} + y^{2} + z^{2}}{2}$
and
$\frac{x^{7} + y^{7} + z^{7}}{7} = \frac{x^{2} + y^{2} + z^{2}}{2} \times \frac{x^{5} + y^{5} + z^{5}}{5}$
I thought this was a fun problem to tackle, but I've been stuck on this for hours, and I can't seem to get anywhere. I've tried expanding, but no matter what I do, I just get a huge mess.
I also cannot find how to incorporate the initial assumption ( $x + y + z = 0$ ) anywhere. Have I missed something?

Lorena Lester 2022-07-16

How did they get 2n + 10 here?
Daniel can encode a paper in $5$ hours and together he and Dennis can encode it in $2$ hours, how long would it take Dennis to encode the same paper alone?
$\frac{1}{5} + \frac{1}{n} = \frac{1}{2}$
$2 n + 10 = 5 n$
$10 = 3 n$
$n = \frac{10}{3}$
$n = 3 \frac{1}{3}$

beatricalwu 2022-07-16

Use partial fractions to solve the integral
\int \frac{x^2-x+20}{x^3+5x}dx

EnvivyEvoxys6 2022-07-15

Prove that if $\frac{x + y}{3 a - b} = \frac{y + z}{3 b - c} = \frac{z + x}{3 c - a}$ then $\frac{x + y + z}{a + b + c} = \frac{a x + b y + c z}{a^{2} + b^{2} + c^{2}}$
I tried to prove this in many ways. First, I tried to multiply the first equality by $(3 a - b) (3 b - c) (3 c - a)$ , but then it seems too complicated. Is there any easy method?

Augustus Acevedo 2022-07-15

Can you express this in form $\frac{1}{a} + \frac{1}{b}$
Can you express the fraction $\frac{1949}{1999}$ in the form $\frac{1}{a} + \frac{1}{b}$ ? Give reasons supporting your answer.
I think the only way to do this is keep trying numbers but then I will never get the answer. I cry every time.

desertiev5 2022-07-15

Find original cost based on fractional purchase
How would I go about finding the original cost of bitcoin knowing that $20 purchased .0531401 of bitcoin? I would like to know what the cost of 1 bitcoin was at the time of purchase?
20 / X = .0531401

lilmoore11p8 2022-07-15

Is $n^{\log c} = c^{\log n}$ true?
If so, please explain.

Caleb Proctor 2022-07-15

Can the powers of 2 in the denominators of a fraction sum be used to find a contradiction?
Let $a_{1}, a_{2}, b_{1}, b_{2}, c_{1}, c_{2}, d_{1}, d_{2}, e_{1}, e_{2}$ be odd integers, with
$gcd (a_{1}, a_{2}) = gcd (b_{1}, b_{2}) = gcd (c_{1}, c_{2}) = gcd (d_{1}, d_{2}) = gcd (e_{1}, e_{2}) = 1,$
and assume $n \geq 3$ and $m \geq 1$ are integers such that
$\begin{matrix} (⋆) & \frac{a_{1}}{2^{3 (n - 1) m} a_{2}} + \frac{b_{1}}{2^{3 ((n - 1) m + 1)} b_{2}} + \frac{c_{1}}{2^{3 ((n - 1) m + 1)} c_{2}} + \frac{d_{1}}{2^{3 n m} d_{2}} + \frac{e_{1}}{2^{3 n m} e_{2}} = 1. \end{matrix}$
Is there a way to prove, using only the powers of 2 in the set of denominators, that (⋆) is impossible?

kolutastmr 2022-07-15

Simplifying a fraction with radicals
Simplify the following fraction as much as possible
$\frac{(8 + 2 \sqrt{15})^{3 / 2} + (8 - 2 \sqrt{15})^{3 / 2}}{(12 + 2 \sqrt{35})^{3 / 2} - (12 - 2 \sqrt{35})^{3 / 2}}$
This problem is just driving me insane. I've tried so many different approaches but it keeps becoming a mess for me
I worked with conjugates but that won't help. The first step would be to get rid of the exponent.
Any help on this problem?

DIAMMIBENVERMk1 2022-07-14

$\frac{x}{y} \geq \frac{a_{1}}{b_{1}} \geq \frac{a_{2}}{b_{2}}$ and $b_{1} \leq b_{2} ⟹ \frac{x + a_{1}}{y + b_{1}} \geq \frac{x + a_{2}}{y + b_{2}}$

Raul Walker 2022-07-14

How do you solve a logarithm with a non-integer base?
How would one calculate the log of a number where the base isn't an integer (in particular, an irrational number)? For example:
${0.5}^{x} = 8 (where x = - 3)$
$\log_{0.5} 8 = - 3$
How would you solve this, and how would this work for an irrational base (like $\sqrt{2}$ )?

Bruno Pittman 2022-07-14

Simplifying $\csc (\frac{5 π}{12})$
I got to the point $\frac{2 \sqrt{2}}{\sqrt{3} + 1}$ but I have to get to $\sqrt{6} - \sqrt{2}}$
Could someone guide me the way?

Kolten Conrad 2022-07-14

How do I work out $\frac{10 \sqrt{3} 0}{5 \sqrt{5}}$ $?$
I am told the answer is $10 \sqrt{6}$ but I keep getting $2 \sqrt{6}$ . How do I get $10 \sqrt{6}$ as the answer $?$

Leila Jennings 2022-07-14

Why $\frac{a}{b} = \frac{c}{d} ⟹ \frac{a + c}{b + d} = \frac{a}{b} = \frac{c}{d}$ ?
In geometry class, this property was quickly introduced:
$\frac{a}{b} = \frac{c}{d} ⟹ \frac{a + c}{b + d} = \frac{a}{b} = \frac{c}{d}$
It usually avoids some boring quadratic equations, so it's useful but I don't really get it. Seems trivial but I can't prove.
How to demonstrate it?

Hayley Bernard 2022-07-14

For $a$ , $b$ , $c$ the sides of a triangle, prove $| \frac{a}{b} - \frac{b}{a} + \frac{b}{c} - \frac{c}{b} + \frac{c}{a} - \frac{a}{c} | < 1$
I initially thought of changing it in terms of sine using the law of sines, which gives me
$| \frac{\sin A}{\sin B} - \frac{\sin B}{\sin A} + \frac{\sin B}{\sin C} - \frac{\sin C}{\sin B} + \frac{\sin C}{\sin A} - \frac{\sin A}{\sin C} | < 1$
However, from there I'm not sure how to proceed. Any tips?

Pattab 2022-07-14

How to find the value of $\sum_{k = 1}^{\infty} (\frac{1}{9})^{k}$ using partial sums?
So I was trying to prove an infinite sum by looking at the partial sum, when I ran into a problem.
Consider:
$\sum_{k = 1}^{n} {(\frac{1}{9})}^{k} = \frac{1}{8} 9^{- n} (9^{n} - 1)$
but as there are no solutions to $9^{- n} (9^{n} - 1) = 1$ , is it possible to prove that
$\sum_{k = 1}^{\infty} {(\frac{1}{9})}^{k} = \frac{1}{8}$
by the partial sum?

Riya Hansen 2022-07-14

Most students asked why is that $\frac{a}{b} \div \frac{c}{d} = \frac{a d}{b c}$
I just told them: inverse the second fraction and multiply. Why? They ask me. I have no idea.
Any logical answers to them kids?
These day teachers just told students at secondary to memorise and no why is allowed in lessons. I think that is wrong. Putting people of studying maths.

kramberol 2022-07-14

How to take the Limits of Logs
$\frac{\log (n!)}{\log (n^{n})}$
as $n \to \infty$
I believe you have to remove the log raising it to their base. Is this correct ?
Thanks.\

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