🧮 algebra

1. **Stating the problem:** We are given three consecutive terms of an exponential sequence: $$10 \cdot \binom{n}{6} / 7, \binom{b}{7}, \binom{n}{8}$$

1. **Énoncé du problème :** Résoudre l'équation logarithmique $$\ln(x+1) - \ln(x-3) = \ln(6x - 9)$$.

1. **Stating the problem:** We need to solve the equation $$t = \frac{301.5}{v \cos 25^\circ}$$ for $v$ given a value for $t$.

1. **State the problem:** Simplify the expression $$2x \sqrt[3]{16x^4} + \sqrt[3]{375x^8} - 2x^2 \sqrt[3]{54x}$$. 2. **Recall the cube root and exponent rules:**

1. **State the problem:** Solve the equation $7 + 9x - 18 = 1$ for $x$. 2. **Simplify the left side:** Combine like terms $7$ and $-18$.

1. **State the problem:** Simplify the expression $$\frac{2x^2 - 8}{4x}$$. 2. **Recall the formula and rules:** To simplify a rational expression, factor numerator and denominator

1. **State the problem:** Simplify the expression $y\sqrt{20} + \sqrt{80y^2}$ assuming all variables are positive. 2. **Recall the rules:**

1. **Problem:** Sketch the graph of the function $y = -2(x - 5)^2 + 8$. 2. **Formula and rules:** This is a quadratic function in vertex form $y = a(x-h)^2 + k$, where $(h,k)$ is t

1. **Stating the problem:** We are given two quadratic functions and asked to analyze their properties including vertex, optimal value, axis of symmetry, zeros, y-intercept, direct

1. Énonçons le problème : Trouver le reste $r(X) = cX + d$ de la division du polynôme $P_m(X) = 2^m - 1^m - 1$ par $X^2 - 3X + 2 = (X-1)(X-2)$. 2. Rappel : Le reste d'une division

1. **State the problem:** Given roots $a$ and $b$ of an equation with conditions $a^2+b^2=21$ and $a+b=-3$, find the equation whose roots are $\frac{a}{b-2}$ and $\frac{b}{a-2}$. 2

1. **State the problem:** A store sold 150 apples in total, and 60 of these were green apples. We need to find what percentage of the apples sold were green. 2. **Formula used:** T

1. **State the problem:** Solve the cubic equation $$2x^3 + 2x^2 - 1040 = 0$$ for $x$. 2. **Write the equation:** $$2x^3 + 2x^2 - 1040 = 0$$

1. المشكلة: نريد معرفة أشكال معادلة الخط المستقيم المختلفة. 2. أشكال معادلة الخط المستقيم:

1. **State the problem:** Solve the equation $2^7 + 2^2 - 1340 = 0$. 2. **Recall the powers of 2:**

1. The problem involves understanding the expressions inside the logarithms and products: $$L_{\Lambda(1-\log_{\lambda}(1+\frac{\prod(\lambda^{1-\frac{\mu_1}{\Lambda}}-1)}{\lambda-

1. **State the problem:** Solve the equation $$e^2 + 5w = 12$$ for $w$. 2. **Identify the formula and rules:** This is a linear equation in $w$. To isolate $w$, subtract $e^2$ from

1. **Problem:** Solve the system $$\begin{cases} x \cos t - y \sin t = a \\ x \sin t + y \cos t = b \end{cases}$$

1. **State the problem:** We want to find what number divides the sum $4^{41} + 4^{42} + 4^{43}$ and check if it is divisible by 7. 2. **Rewrite the sum:** Factor out the smallest

1. **State the problem:** We want to find a number by which the sum $4^{41} + 4^{42} + 4^{43}$ is divisible. 2. **Rewrite the expression:** Notice that $4^{42} = 4^{41} \times 4$ a

1. Let's consider a simple algebra problem: Solve for $x$ in the equation $2x + 3 = 7$. 2. The formula used here is to isolate $x$ by performing inverse operations. We want to get