🧮 algebra

1. **State the problem:** Find the domain of the function $$f(x) = \frac{1}{\frac{12}{x-5} - 2}$$. 2. **Recall domain rules:** The domain of a function includes all real numbers ex

1. The problem asks to determine values for $A$ and $B$ such that the system of equations $$\begin{cases} 12x + Ay = 8 \\ Ax + By = 4 \end{cases}$$

1. **Problem:** Determine which point lies on the line given by the equation $$2x - 3y = 7$$. 2. **Formula and rule:** To check if a point $ (x, y) $ lies on the line, substitute t

1. **Problem:** Determine which point lies on the line given by the equation $$2x - 3y = 7$$. 2. **Formula and rule:** To check if a point $ (x, y) $ lies on a line, substitute the

1. **State the problem:** Solve the equation $5 - 6(2x + 2) = -2(6x + 5) + 4$ using the method of drawing a vertical line under the equal sign to keep both sides balanced. 2. **Dis

1. **State the problem:** Simplify the expression $$\frac{(8x^6y^{-3})^{\frac{1}{3}}}{\sqrt{16x^8}}$$. 2. **Recall the rules:**

1. **State the problem:** Solve the equation $5 - 6(2x + 2) = -2(6x + 5) + 4$ using the method of drawing a vertical line under the equal sign to keep both sides balanced. 2. **Dis

1. **State the problem:** We need to find the value of $a$ in the equation $$\left(x^{\frac{1}{2}}\right)^3 \sqrt{x} = x^a$$ given that $x > 0$. 2. **Recall exponent rules:**

1. The problem is to simplify the expression $$(243x^4)^{\frac{4}{5}}$$. 2. We use the power of a product rule: $$(ab)^m = a^m b^m$$.

1. **State the problem:** Evaluate the expression $125^{-\frac{2}{3}}$. 2. **Recall the rule for negative and fractional exponents:**

1. **State the problem:** We need to find the first four terms of a sequence starting at $\frac{5}{6}$ where each term is obtained by adding $\frac{2}{3}$ to the previous term. 2.

1. The problem asks for the first four terms of a sequence starting at 5 where each term is obtained by adding 3 to the previous term. 2. This is an arithmetic sequence where the f

1. **State the problem:** Simplify the expression $$\frac{(-2)^3 \times (-2)^4}{(-2)^2 \times (-2)^5} + \frac{(-1)^0 \times (-1)^3}{(-1)^4 \times (-1)^x} \cdot \frac{(-1)^3 \cdot (

1. **State the problem:** Simplify the expression $$\frac{(12 m^5 n^{-2})(5 m^{-11} n^6)}{15 m^3 n^{-4}}$$. 2. **Multiply the numerator terms:**

1. The problem is to evaluate $\log_3(9)$.\n\n2. Recall the definition of logarithm: $\log_b(a) = c$ means $b^c = a$. Here, $b=3$, $a=9$, and we want to find $c$.\n\n3. We want to

1. **State the problem:** Simplify the complex fraction $$\frac{-1/2}{1/6}$$. 2. **Recall the rule for dividing fractions:** Dividing by a fraction is the same as multiplying by it

1. The problem asks: What does it mean to express a number or expression in exponential form? 2. Expressing in exponential form means writing a number as a base raised to an expone

1. **Problem Statement:** Mary makes $n$ bracelets each week and sells them for $19 - n$ dollars each.

1. The problem asks us to compare the fraction $-\frac{11}{8}$ with the decimal $-1.37$ to determine which is greater. 2. To compare these two numbers, it is easiest to convert the

1. **State the problem:** Determine if $\frac{11}{8}$ is greater than or less than $-1.37$. 2. **Recall the values:**

1. **Problem 2:** Determine approximately how long the weight is higher than 4 m based on the graph of height $h$ versus time $t$. 2. From the graph description, the weight starts