$\mathrm{NH}_3$ in water reacts to form $\mathrm{NH}_4{ }^{+}$in the equilibrium reaction:

$$\mathrm{NH}_3+\mathrm{H}_2 \mathrm{O} \rightleftharpoons \mathrm{NH}_4^{+}+\mathrm{OH}^{-}$$

(a) Is $\mathrm{NH}_3$ a strong or weak base?

(b) Explain why the $\mathrm{pH}$ of a $\mathrm{NH}_3$ solution is lower than that of a $\mathrm{NaOH}$ solution:

Convert the angle $73.415^{\circ}$ to degrees, minutes and seconds.

Find whether the relationship is proportional.

For the following exercise, convert the quadratic functions to vertex form by completing the square. Identify the vertex and the axis of symmetry.

$$u(z)=-3 z^2+9 z-2$$

The product $\frac{x^2-4}{x^2-3 x-4} \cdot \frac{x-4}{x+2}$ in simplest form is $\frac{x-2}{x+1}$, as shown:

$\frac{x^2-4}{x^2-3 x-4} \cdot \frac{x-4}{x+2}=\frac{(x+2)(x-2)}{(x+1)(x-4)} \cdot \frac{(x-4)}{(x+2)}=\frac{x-2}{x+1}$. When 4 is substituted for $x$ in $\frac{x-2}{x+1}$, the value is $\frac{2}{5}$. Explain why the value of $\frac{x^2-4}{x^2-3 x-4} \cdot \frac{x-4}{x+2}$ is not $\frac{2}{5}$ when $x=4$.

a) Find a recurrence relation for the number of steps needed to solve the Tower of Hanoi puzzle.

b) Show how this recurrence relation can be solved using iteration.

Multiply or divide as indicated. Simplify and leave the numerator and denominator in your answer in factored form. Assume all denominators are nonzero. $\frac { x ^ { 2 } - 1 } { 3 x } \div \frac { 7 x - 7 } { x ^ { 2 } + x }$

The net force along the linear path of a particle of mass $480 \mathrm{~g}$ has been measured at $10.0-\mathrm{~cm}$ intervals, starting at $x=0.0$, to be $26.0,28.5,28.8,29.6,32.8,40.1,46.6,42.2$, $48.8,52.6,55.8,60.2,60.6,58.2,53.7,50.3,45.6,45.2,43.2,38.9$, $35.1,30.8,27.2,21.0,22.2$, and $18.6$, all in newtons. Find the total work done on the particle over this entire range.

A titration of 25.0 mL of a solution of the weak base aniline, ${C}_6 {H}_5 {NH}_2$, requires 25.67 mL of 0.175 M HCl to reach the equivalence point.

$${C}_6 {H}_5 {NH}_2({aq})+{H}_3 {O}^{+}({aq}) \rightleftarrows {{C}_6 {H}_5 {NH}_3{ }^{+}({aq})+{H}_2 {O}(\ell)}{}$$

(a) What was the concentration of aniline in the original solution?

(b) Determine the concentrations of ${H}_3 {O}^{+}, {OH}^{-}$, and ${C}_6 {H}_5 {NH}_3{ }^{+}$at the equivalence point.

(c) Determine the pH of the solution at the equivalence point.

Determine whether A is diagonalizable and, if so, find an invertible matrix P and a diagonal matrix D such that

$$P ^ { - 1 } A P = D.$$

$$A = \left[ \begin{array} { l l } { - 3 } & { 4 } \\ { - 1 } & { 1 } \end{array} \right]$$

A 0.15 M solution of a weak acid is found to be 1.3% ionized. What is its Ka?

Use symmetry to sketch the graph of the polar equation. Use a graphing utility to verify your graph. $r=4(1+\sin \theta)$

Write an equation to represent the given relationship among integers. The product of two consecutive even integers is 168.

Use complex numbers to find an invertible matrix $P$ and a diagonal matrix $D$ such that $A=PDP^{-1}$.

$$\begin{bmatrix}0&&-1&&1\\3&&3&&-2\\2&&1&&1\end{bmatrix}$$