Simplify 1024/384 efficiently by recognizing powers of two. What is the simplified fraction?

For which integers n is the equality (-2)ⁿ = -2ⁿ true? Select the best answer and justify by parity reasoning.

A projectile's height is modeled by h(t) = -16t² + vt + s. If v = 64 ft/s and s = 5 ft, compute the height h(2). Use the appropriate mathematical methods.

A traveler has b dollars. They spend 120 dollars on a hotel and then use one-third of the remaining money for food. Represent the amount spent on food as an algebraic expression in terms of b.

Translate the sentence and solve the resulting quadratic: "The product of a number and the difference of seven and that number is ten."

A baker packs 8 muffins and 5 cookies in each box. A customer orders 12 identical boxes and also buys 7 extra cookies. Write an expression for the total number of items.

A small business charges setup fee S and per-item cost C. Their combined weekly cost for two product runs is modeled by: 2(S + 3C) - (S - 4C) + 5. Simplify this expression to combine like terms and express the weekly cost in terms of S and C.

Evaluate the following student's solution to 3x − 4 = 11: they added 4 to both sides to get 3x = 7, then divided by 3 and concluded x = 7/3. Determine whether the steps and final answer are correct, and if not provide the correct solution.

Solve the equation (1/2)x + 1/3 = 7/6.

Solve for $x$: $\frac{2}{5}x-\frac{3}{4}=\frac{1}{10}x+\frac{7}{8}$

Solve the inequality for $x$.

Solve the system by substitution: 3x + 4y = 18 and y = 2x - 3. Provide exact values.

Given the formula for compound interest A = P(1 + r/n)^nt, isolate r symbolically (write r in terms of A, P, n, and t).

A jacket priced at $\$250$ is discounted by $30\%$. A $7\%$ sales tax is then applied to the discounted price. What is the final price?

A chemist needs to create 14 liters of a 50% acidic solution. They have two different acidic solutions available to use each with percentages of 40% and 70% respectively. If $x$ represents the number of liters for the 40% solution, how many liters of each solution should they mix together in order to achieve their desired acidic solution?

A table shows that when $x=0$, $y=3$, and when $x=2$, $y=7$. Assuming $y$ depends linearly on $x$, find the linear equation $y=mx+b$, and then compute $y$ when $x=4$.

Given two points observed on a graph, $(2,-1)$ and $(5,8)$, derive the equation of the line in standard form $ax+by=c$ with integer coefficients.

Find the $x$- and $y$-intercepts of the line given by $3x-y=12$.

A wheelchair ramp must rise 2 feet for every 12 feet horizontally to meet a building entrance. Starting from ground-level point (0,0), write the slope and find the coordinates of the point at the top of a 36-foot section of ramp, then state the slope-intercept equation of the ramp line.

Convert $y-\frac{3}{2}=\frac{4}{3}(x+\frac{1}{3})$ into slope-intercept form $y=mx+b$.