🧮 algebra

1. **Problem 1:** Simplify the expression $\frac{a^2 - b^2}{a + b}$. 2. **Formula and rules:** Recall the difference of squares factorization:

1. **Stating the problem:** Identify the correct expansion of the binomial expressions given. 2. **Formula used:** The binomial expansion formulas are:

1. **Problem:** Identify the correct expansion of the expression $(a - b)^2$. 2. **Formula:** The expansion of $(a - b)^2$ is given by:

1. The problem is to find the value of the expression $$3 \log_3 27 + \ln 1$$. 2. Recall the logarithm rules: $$\log_a a^x = x$$ and the natural logarithm rule: $$\ln 1 = 0$$.

1. The problem asks to evaluate $\frac{\log_k 32}{\log_k 2}$ for $k > 0$. 2. We use the change of base formula: $\frac{\log_k a}{\log_k b} = \log_b a$.

1. The problem is to evaluate the expression $e^a \times 4$ where $e$ is the base of the natural logarithm and $a$ is a variable. 2. The formula used here is the exponential functi

1. **State the problem:** Simplify the expression $$\frac{7x}{12} + \frac{x-2}{6} - \frac{x^2}{3}$$. 2. **Find a common denominator:** The denominators are 12, 6, and 3. The least

1. **State the problem:** Find the x-intercept and y-intercept of the line shown on the graph. 2. **Recall definitions:**

1. **State the problem:** Find the intercepts of the line given by the equation $$y = 5x - 13$$. 2. **Recall the intercept definitions:**

1. **State the problem:** We need to fence a rectangular garden with dimensions 3 m by 6 m without cutting or bending any fence pieces. The fence pieces come in lengths 0.6 m, 1.8

1. **State the problem:** Maria has a rectangular board measuring 150 cm by 30 cm. She cuts it into four smaller rectangular pieces, each with the same length-to-width ratio as the

1. **State the problem:** Find the value of the summation $$\sum_{i=1}^{50} (2i + 1)$$. 2. **Recall the formula:** We know the formula for the sum of the first $n$ integers is $$\s

1. **State the problem:** Find the value of $c$ in the equation $2x + 5y = c$ for the line passing through points $(-1, 2)$ and $(4, 0)$. 2. **Formula and rule:** The equation of a

1. **Problem:** The third term of an arithmetic progression (AP) is 9, and the seventh term is 49. Calculate the thirteenth term. 2. **Formula:** The $n$th term of an AP is given b

1. **State the problem:** Simplify the expression $$\frac{x^2 - 1}{x^3 + 8} \times \frac{x^9 - x^8 - 6x^7}{x^5 + x^4} \div \frac{x^3 - 4x^2 + x + 6}{x^4 - 16}$$

1. **State the problem:** Solve the system of equations using the elimination method: $$3x + 2y = 6$$

1. State the problem: Create a question involving logarithmic equations using the laws of logarithms. 2. Use the logarithmic laws:

1. El problema es resolver la ecuación dada (aunque no se especifica, asumimos que se refiere a resolver una ecuación algebraica simple). 2. Para resolver una ecuación, usamos la r

1. **Problema:** Calcule el producto y la división de expresiones algebraicas dadas. 2. **Fórmulas y reglas importantes:**

1. **Stating the problem:** We are given the equation:

1. State the problem: Solve for $a$ in $a^3+a^2=36$. 2. Set up the equation.