🧮 algebra

1. The problem is to simplify and understand the expression $$b_n = e^n + 2n(1 + 3e^n)^2$$. 2. First, expand the squared term $$(1 + 3e^n)^2$$ using the binomial formula $$(a+b)^2

1. The problem states that the volume $v$ in the container depends on the number of cubes $x$ according to the equation: $$v = 8 + 4x$$

1. The problem asks for the slope of the line that relates the number of cubes dropped into the container ($x$) to the volume of water in the container ($V$). 2. From the graph des

1. Problem 15: Solve the equation $a + 80 = 8a$. Step 1: Subtract $a$ from both sides to isolate terms with $a$ on one side:

1. The problem is to simplify the expression $\log_4 9 + \log_4 21 - \log_4 7$.\n\n2. Use the logarithm property: $\log_b a + \log_b c = \log_b (a \times c)$ and $\log_b a - \log_b

1. Решити систем линеарних једначина: $$\begin{cases}-x + 2y - z = 8 \\ 2x - 4y + 3z = 6 \\ x + 2y - 2z = 0 \end{cases}$$

1. **State the problem:** Solve the equation $$\frac{4}{x-2} = \frac{3}{x^2-4} - \frac{1}{4}$$ for $x$. 2. **Factor the denominator:** Note that $$x^2 - 4 = (x-2)(x+2)$$.

1. The problem is to evaluate $\frac{\log_3 1}{243}$.\n\n2. Recall that $\log_3 1$ means the power to which 3 must be raised to get 1. Since $3^0 = 1$, we have $\log_3 1 = 0$.\n\n3

1. Problem: Simplify the expression $9y^2 + x - 3y^2 + 3x$. 2. Step 1: Identify and group like terms by type: group the $y^2$ terms together and the $x$ terms together.

1. The problem is to evaluate $\log\left(\frac{2}{3}\right)$.\n\n2. Recall the logarithm property: $\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$.\n\n3. Applying this property,

1. The problem is to simplify the expression $\log_A \frac{2}{3B}$. 2. Recall the logarithm property: $\log_A \frac{x}{y} = \log_A x - \log_A y$.

1. **State the problem:** Find all values of $m$ such that the inequalities hold for every real number $x$: 1. $m x (x - 2) < m^2 x^2 + 2$

1. Le problème est de résoudre un système d'équations linéaires en utilisant la méthode de Gauss-Jordan. 2. La méthode de Gauss-Jordan consiste à transformer la matrice augmentée d

1. The problem asks for the two answers, but no specific equation or context is given. 2. To find two answers, typically we solve a quadratic equation or a problem with two solutio

1. The problem is to find the vertical asymptote(s) of the curve given by the function $$y=\frac{x^2-4}{x-5}$$. 2. Vertical asymptotes occur where the denominator of a rational fun

1. **State the problem:** Simplify the expression $$A = \frac{(a^5 b^{-6})^{-2} (-a)^4 b^{-10}}{b^{10} (a^{-7})^3}$$. 2. **Simplify each part:**

1. The problem is to solve the inequality $|6x+2| > -2$. 2. Recall that the absolute value $|A|$ of any expression $A$ is always greater than or equal to 0.

1. Stating the problem: Solve the equation $$3\left(2x - 2\frac{2}{3}\right) = -42$$. 2. Convert the mixed number to an improper fraction: $$2\frac{2}{3} = \frac{8}{3}$$.

1. The problem is to solve the inequality $|6x + 2| \leq 0$. 2. Recall that the absolute value $|A|$ is always non-negative, and $|A| = 0$ if and only if $A = 0$.

1. نبدأ بحل المتباينة الأولى: س - 3 < 2 < س 2. نحل الجزء الأول من المتباينة: س - 3 < 2

1. مسئله را بیان می‌کنیم: نامساوی ترکیبی $$-3 < س - 2 < 7$$ را حل کنیم و همچنین نامساوی $$2 س > -3$$ را بررسی کنیم. 2. ابتدا نامساوی اول را به دو نامساوی جداگانه تقسیم می‌کنیم: