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

Evaluate the expression −2⁴ + (−2)⁴ and explain the difference created by parentheses.

Evaluate (5 − 2³)/3 + (1/6) · 12. Break the problem into sub-steps, then focus on solving the exponent and fractions, and then determine the exact final value.

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."

Expand and simplify 2(3a - 4b + 5c - 6).

Distribute and simplify this expression with fractions: (1/2)(4x - 6) - (1/3)(3x + 9).

Solve 4x + 2 = 10, verify your solution by substitution, and express the final answer in set notation.

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

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

A storage company charges a one-time setup fee of $\mathrm{\backslash (}35$ and a monthly fee of $\mathrm{\backslash )}20$. You want the total cost to be strictly less than $\$200$. Let $m$ be the number of months. Set up the inequality and determine the maximum whole number of full months you can rent storage.

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?

Which algebraic translation correctly represents: "There are eight more nickels than dimes," if d represents the number of dimes and n represents the number of nickels?

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 both intercepts for the equation $-3x+6y=12$.

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 $y=mx+b$.

Find the equation of the line passing through (1/3, 2) and (4/3, −1). Provide the point-slope form using the first point and then convert to slope-intercept form.

Given the system of constraints $x\ge 0$, $y\ge 0$, $x+2y\le 8$ and $y\ge x-1$, determine whether the point $(3,2)$ is in the feasible region that satisfies all constraints.

Find the domain of $y=\sqrt{\frac{x+1}{x-2}}$. Express your answer in interval notation.

A linear function $f$ is known to satisfy $f\left(1\right)=3$ and $f\left(3\right)=7$. Find the constants $a$ and $b$ so that $f\left(x\right)=ax+b$.

Solve the system: $y=\frac{2}{3}x-1$ and $3x+2y=7$. Provide the intersection point.

Solve the system using substitution: { y = (3/4)x - 2 ; (1/2)x + (3/2)y = 6 }

Solve the system: $-2x+7y=1$ and $4x-3y=11$. Provide the final ordered pair.

A company makes two products: $A$ and $B$. Each product $A$ gives \(30 in profit and uses 2 hours of labor; each product $B$ gives \)50 in profit and uses 3 hours of labor. The weekly labor available is at most 60 hours per week. They must make at least 5 units of product $A$. There is no minimum for $B$. Set up a system of inequalities and determine which corner of the feasible region (integer or fractional that is allowed) that will maximizes the profit: $P=30A+50B$. Show all calculations.