Collecting all Math questions and answers in one place

$\frac{20b}{{\left(4b^3\right)}^3}$

Why are quadratics factored into 2 brackets?
${\displaystyle 2x^2+8x+6}$
${\displaystyle 2x(x+4+3\times \frac{1}{x})}$ or ${\displaystyle 2x(x+4+\frac{3}{x})?}$

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Why do we integrate along the whole length while finding the gravitational force between a object of mass (m) and a rod of length L and mass M? Can't we simply use $F=\frac{GmM}{r^2}?$?

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Simplify the expression $1-{\sin }^{2}x$

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Expressing $\ln <!--\; \; -->\sqrt[3]{54}$ in terms of $\ln <!--\; \; -->2$ and/or $\ln <!--\; \; -->3$
I know that $\sqrt[3]{54}={54}^{1/3}$ but otherwise I don't know how to address these types of problems. How do I solve this, and is there a general way of tackling these problems? Like, if I had $\ln <!--\; \; -->.75$ or $\ln <!--\; \; -->\frac{8}{9}$, for example. These types of problems that I have are all different and I don't understand a general way of looking at them.
thanks.

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Solve the simultaneous Linear equations using matrix inverse Method.
8x+3y=2
6x+2y=4

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Discrete Mathematics:
Find ${\displaystyle \sum _{i=1}^{i=4}\prod _{j=2}^{j=3}(i+\lceil j+0.5\rceil )}$

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Probability vs Confidence
My notes on confidence give this question:
An investigator is interested in the amount of time internet users spend watching TV a week. He assumes ${\displaystyle \sigma =3.5}$ hours and samples ${\displaystyle n=50}$ users and takes the sample mean to estimate the population mean ${\displaystyle \mu }$.
Since ${\displaystyle n=50}$ is large we know that ${\displaystyle \frac{\bar{X}-\mu }{\frac{\sigma }{\sqrt{n}}}}$ approximates the Standard Normal. So, with probability ${\displaystyle \alpha =0.99}$, the maximum error of estimate is ${\displaystyle E=z_{\frac{\alpha }{2}}\times \frac{\sigma }{\sqrt{n}}\approx 1.27}$ hours.
The investigator collects that data and obtain ${\displaystyle \bar{X}=11.5}$ hours. Can he still assert with 99% probability that the error is at most 1.27 hours?
With the answer that:
No he cannot, because the probability describes the method/estimator, not the result. We say that "we conclude with 99% confidence that the error does not exceed 1.27 hours."
I am confused. What is this difference between probability and confidence? Is it related to confidence intervals? Is there an intuitive explanation for the difference?

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The volume of a cube is increasing at a rate of 1200 cm^3/min at the moment the lengths of the sides are 20 cm. How fast are the lengths of the sides increasing at the moment?

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