Master Fraction Math Problems with Expert Help

How do you calculate volume with cubes of fraction lengths?
I keep having trouble with one exercise. The questions are like this: how many $1 / 4$ cm cubes does it take to fill a prism if the prism is $1$ cm $\times$ $7 / 4$ cm $\times$ $2$ cm?
I've tried converting all the side lengths to multiples(?) of $1 / 4$ , such as $4 / 4 \times 7 / 4 \times 8 / 4$ and multiplying but I still get the wrong answer. How do I calculate this?

tuzkutimonq4 2022-09-08

Proving that $\frac{1}{3 x^{2} + 1} + \frac{1}{3 y^{2} + 1} + \frac{1}{3 z^{2} + 1} \geq \frac{3}{16}$
Let $x, y, z \geq 1$ and $x + y + z = 6$ . Then
$\frac{1}{3 x^{2} + 1} + \frac{1}{3 y^{2} + 1} + \frac{1}{3 z^{2} + 1} \geq \frac{3}{16} .$
I tried to use Cauchy- Schwartz inequality but it doesn't work.

Slovenujozk 2022-09-08

Find $(2.3 \times 104) \div (9.2 \times 106)$ by using a calculator. Give your answer in scientific notation and round to one decimal place.

Lisantiom 2022-09-07

Division of convex functions
I need your expertise in understanding the following:
Let $n \in N$ , $x_{i} \in R$ for every $i \in [n]$ and let $a \in R_{+}$
What can be said about the following in term of convexity (let $j$ be any arbitrary integer such that $i \in [n]$
$\frac{\frac{a^{2}}{2 n} + max {0, 1 - x_{i}}}{\frac{a^{2}}{2} + \sum_{j \in [n]} max {0, 1 - x_{j}}}$
I am asking this since, it's easy to see that both the denominator and nominator are convex (it resembles the objective function of SVM), however is this fraction convex, or quasi-convex, concave, etc... ?
Please advise and thanks in advance.
P.s. A more advanced question would be, what can be said on the fraction of two convex function in general?

abkapseln87 2022-09-07

Simplify fraction with a $\pm$ value numerator
What are the steps to simplify by $2$ as shown? I am not clear on how this is does while still maintaining the $\pm$
$\frac{6 \pm 2 \sqrt{6}}{- 6}$
To:
$\frac{3 \pm \sqrt{6}}{- 3}$

Makayla Reilly 2022-09-07

Show that $(a + 2 b + \frac{2}{a + 1}) (b + 2 a + \frac{2}{b + 1}) ⩾ 16$ if $a b ⩾ 1$
$a$ , $b$ are positive real numbers.
Should I use AM-GM inequality? If yes, where?

Ciolan3u 2022-09-07

Solving a Quadratic Inequality
Solve the inequality:
$\frac{x - 1}{x^{2} - 4 x + 3} < 1.$
I cancelled the $x - 1$ from the numerator and the denominator and I obtained:
$\frac{x - 4}{x - 3} .$
Thus, the answer that I got was
$x \in (3, 4) .$
But according to WolframAlpha answer should be
$x \in (4, \infty) \cup (1, 3) \cup (- \infty, 1) .$
What is the best method to solve this inequality and where have I gone wrong?

Kirsten Mcintyre 2022-09-07

Prove without using of calculator, that $\sin (40^{\circ}) < \sqrt{\frac{3}{7}}$
My attempt.
Since
$\sin (40^{\circ}) = 2 \sin (20^{\circ}) \cos (20^{\circ}) < 2 \sin (20^{\circ})$
$= 2 \sin (60^{\circ} - 40^{\circ}) = \sqrt{3} \cos (40^{\circ}) - \sin (40^{\circ}),$
$2 \sin (40^{\circ}) < \sqrt{3} \cos (40^{\circ}) .$
Hence,
$4 \sin^{2} (40^{\circ}) < 3 \cos^{2} (40^{\circ}) = 3 (1 - \sin^{2} (40^{\circ}))$
$7 \sin^{2} (40^{\circ}) < 3$
$\sin (40^{\circ}) < \sqrt{\frac{3}{7}}$

Zackary Duffy 2022-09-07

What is $\frac{3}{4}$ divided by $7 \frac{1}{4}$ ?

ezelsbankuk 2022-09-06

Simplify the fractions
First, I apologize for the title, but if I write my question then the characters will be too long.
Here's my question:
Why $(\frac{\log 3}{\log 2} + \frac{2 \log 3}{2 \log 2}) (\frac{2 \log 2}{\log 3} + \frac{\log 2}{2 \log 3}) = (2 \times \frac{\log 3}{\log 2}) (\frac{5}{2} \times \frac{\log 2}{\log 3})$ ?
I have no idea how the left side becomes the right side.

firmezas1 2022-09-06

1 College math competitions ranking
Let $a, b, c, d$ be real numbers which are strictly positive.
Show that
$1 < \frac{a}{a + b + c} + \frac{b}{b + a + d} + \frac{c}{c + a + d} + \frac{d}{d + c + b} < 2.$
I need help please.

kybudmanqm 2022-09-06

Solving the indefinite integral $\int \frac{1}{x^{2} + x + 1} d x$
I want to solve the following indefinite integral: (1)
$\int \frac{1}{x^{2} + x + 1} d x$
completing the square: (2)
$= \int \frac{1}{(x + \frac{1}{2})^{2} + \frac{3}{4}} d x$
Substitution: (3)
$u = \frac{2 x + 1}{\sqrt{3}}$
brings: (4)
$d x = \frac{\sqrt{3}}{2} d u$
(5)
$= \frac{2}{\sqrt{3}} \int \frac{1}{u^{2} + 1} d u$
Then getting standard integral (6)
$= \arctan (u)$
and solving the integral and substitute back is not the problem. But I have a problem understanding the substitution in step (3). I don't know where the $\sqrt{3}$ beneath the fraction line comes from and how the formula ends in the standard integral in (5) on the right side. Could someone explain these steps in a bit more detail?

Milton Anderson 2022-09-06

Proof expression: $\cos (80 °) * \sin (70 °) * \cos (60 °) * \sin (50 °) = \frac{1}{16}$

Presley Esparza 2022-09-05

Rationalize a fraction.
Rationalize the denominator $\frac{1}{2^{\frac{1}{3}} + 3^{\frac{1}{3}} + 4^{\frac{1}{3}}}$ . Is there a short solution for this task ? Thanks in advance.

hazbijav6 2022-09-05

Simple absolute value inequality
Prove: for all $x \neq 0,$ $| x + \frac{1}{x} | > 2$ and $| x + \frac{1}{x} | = 2 ⟺ x = \pm 1$
case 1: $x > 0$
$x + \frac{1}{x} > 2$
$x^{2} - 2 x + 1 > 0$
$x_{1, 2} = \frac{2 + \sqrt{4 - 4}}{2} = 1$
Therefore $x \neq 1$
Case 2: $x < 0$
In the same manner we get
$x^{2} + 2 x + 1 > 0$
and $x \neq - 1$
Now we need that both cases will occur simultaneously (?) so we get $x =\neq \pm 1$ for an equality and $x \neq 0$ for inequality

Parker Pitts 2022-09-05

For $a b c = 1$ prove that $\sum_{cyc} \frac{1}{a + 3} \geq \sum_{cyc} \frac{a}{a^{2} + 3}$
I tried TL, BW, the Vasc's Theorems and more, but without success.
I proved this inequality!
I proved also the hardest version: $\sum_{c y c} \frac{1}{a + 4} \geq \sum_{c y c} \frac{a}{a^{2} + 4}$
Thanks all!

cjortiz141t 2022-09-05

Trigonometric identity of finite terms
Prove that:
$\frac{1}{\cos x + \cos 3 x} + \frac{1}{\cos x + \cos 5 x} + \dots + \frac{1}{\cos x + \cos (2 n + 1) x} = \frac{1}{2} \csc x [\tan (n + 1) x - \tan x]$
I tried to prove this using the regular formulas. But failed. Please help me.

Randall Booker 2022-09-04

Write the ratio: $\frac{2}{9}$ to $\frac{7}{9}$ as a fraction in lowest terms.

Gavyn Whitehead 2022-09-04

Proving Algebraic expression involving a Summation
I'm following a derivation and am stumped by one of the steps:
$\sum_{k = 1}^{13} (1 - \frac{k - 1}{13})^{3} = \frac{1}{13^{3}} \sum_{k = 1}^{13} k^{3}$
I am stumped as to how
$(14 - (k - 1))^{3} = k^{3}$
Any help advice appreciated

Chebenk5 2022-09-04

Express f(f(x)) as a single simplified fraction.
f(x) = x/x+1

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