Rationalize and simplify: $\frac{2+\sqrt{7}}{4-3\sqrt{7}}$

Simplify $\sqrt{-200}$ fully by extracting perfect-square factors.

Rationalize and simplify $\frac{3}{2-5i}$, and express the result in standard form $a+bi$ in lowest terms.

Solve $(2x+3{)}^2=50$ and simplify the answers.

Solve $x^2+\frac{7}{2}x-3=0$ by completing the square. Give the exact roots in the simplest radical form.

Find the complex roots of $x^2+4x+13=0$ and express them in the simplest $a+bi$ form.

Convert the quadratic: y = x² − 6x + 5 into vertex form by completing the square, and identify the vertex.

Which simple graphical procedure determines whether a curve $y=g\left(x\right)$ represents a function of $x$?

Solve $5^{2x-1}=125$ for $x$.

A population grows according to $P\left(t\right)=200\cdot 2^{\frac{t}{3}}$, where $t$ is in months. Solve for $t$ when $P\left(t\right)=1600$.

Expand: ${\log }_{10}\left(20x^2{y}^3\right)$

Which equation corresponds to a circle with center at $(5,-2)$ and passing through the point $(8,2)$?

For the ellipse $\frac{x^2}{36}+\frac{y^2}{25}=1$, find the $y$-coordinates of the points on the ellipse with $x=3$.

Convert the equation $y=2x^2+8x+5$ into vertex form by completing the square, and identify the vertex.

Consider the sequence defined by $a_n=(-1{)}^n\cdot \frac{n^2}{2^n}$ for $n\ge 1$. Which classification best describes the behavior and properties of this sequence?

An arithmetic sequence has $a_7=50$ and $a_1=8$. Write the explicit formula for $a_n$.

A geometric sequence has $a_2=12$ and $a_5=96$. Determine $a_1$ and $r$, and write the general formula for $a_n$.

For an integer $n\ge 3$, simplify the expression $\frac{n!}{(n-3)!}$.