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

Geometric proof of ${\displaystyle \frac{{\sin 60}^{\circ }}{{\sin 40}^{\circ }}=4{\sin 20}^{\circ }{\sin 80}^{\circ }}$

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How do you solve ${\displaystyle x^2+16=0}$ using the quadratic formula?

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How to expand the following notation?
Can someone help me to expand the below equation for $N=2$ and $N=3$?
$f_i=\sum <!--\; \sum \; -->\{e_i:j\le <!--\; \le \; -->i\le <!--\; \le \; -->k\}$ for some $1\le <!--\; \le \; -->j\le <!--\; \le \; -->k\le <!--\; \le \; -->N$.
Here $e_i$ is the unit vector of coordinate i in an N-dimensional space.

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What is the integral of log(z) over the unit circle?
I tried three times, but in the end I am concluding that it equals infinity, after parametrizing, making a substitution and integrating directly (since the Residue Theorem is not applicable, because the unit circle encloses an infinite amount of non-isolated singularities.)
Any ideas are welcome.
Thanks,

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Use a table of integrals with forms involving eu to find the indefinite integral ${\displaystyle \int e^{-4x}\sin 3xdx}$

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$$(-<!--\; -\; -->7n^3+6n^2+5n+1)+(2n^3+5n^2+4n+1)=?$$

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How am I supposed to find the solution to the non-homogeneous ode ?
$\overrightarrow{Y}\text{'}=\left(\begin{array}{cc}-4& 2\\ 2& -1\end{array}\right)\overrightarrow{Y}+\left(\begin{array}{c}{x}^{-1}\\ 2x^{-1}+4\end{array}\right)$

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A garage bag  can hold 64 gallons. Convert to litres.

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