🧮 algebra

1. The problem is to add pairs of fractions and simplify the results. 2. The formula for adding fractions with the same denominator is:

1. **Énoncé du problème :** On considère la fonction $f$ définie sur $[0; +\infty[$ par

1. The problem is to find the cube of a number, which means multiplying the number by itself three times. 2. The formula for the cube of a number $x$ is:

1. **State the problem:** We want to expand and simplify the expression $$(a + b)^4$$. 2. **Formula used:** The binomial theorem states that $$(a + b)^n = \sum_{k=0}^n \binom{n}{k}

1. **State the problem:** Simplify the expression $$\cos\left(2\left(x+2p(x^2+1)\sin(2x)+p\frac{x\cos(2x)}{(x^2+1)^{3/2}}\right)\bigg/\sqrt{\left(x+2p(x^2+1)\sin(2x)+p\frac{x\cos(2

1. Problem: Given the sequence defined by the formula $$a_n=(-1)^n \cdot \frac{n+1}{2}$$ for each natural number $$n \geq 1$$, find the third term $$a_3$$. 2. Formula: The general

1. **Problem statement:** Solve the system of equations graphically:

1. **Nêu bài toán:** Cho cấp số nhân $(u_n)$ với số hạng đầu $u_1 = 5$ và công bội $q = 2$. Tính số hạng thứ 5, tức là $u_5$.

1. **State the problem:** We are given that the sum of two numbers is 95. 2. **Formula used:** If the two numbers are $x$ and $y$, then the sum is given by the equation:

1. نبدأ بفهم السؤال: العبارة التي تريد معرفة على ماذا تقبل القسمة. 2. لتحديد القواسم، نحتاج إلى معرفة العبارة أو التعبير الرياضي الذي تتحدث عنه.

1. Problem: Mamy ciąg geometryczny trzywyrazowy $(12, 6, 2m-1)$ i chcemy określić, czy jest rosnący czy malejący oraz znaleźć wartość $m$. 2. Wzór ciągu geometrycznego: W ciągu geo

1. **State the problem:** Calculate the value of $3^{2024} + 9$. 2. **Recall the properties of exponents:** Note that $9$ can be written as $3^2$ because $9 = 3^2$.

1. The problem is to solve the equation or expression provided by the user. Since no specific equation was given, please provide the exact problem to solve. 2. Generally, to solve

1. **State the problem:** Simplify the expression $$\frac{6y}{x} \div \frac{2}{3xy^2}$$. 2. **Rewrite division as multiplication:** Dividing by a fraction is the same as multiplyin

1. **Simplify** $\frac{6y}{x} \div \frac{2}{3xy^2}$. 2. Division of fractions means multiplying by the reciprocal:

1. Stwierdzenie: "Pierwszy wyraz ciągu (a_n) jest dwa razy większy od trzeciego wyrazu tego ciągu." Ciąg jest określony wzorem $$a_n = (-1)^n \cdot (n-5)$$ dla $$n \geq 1$$.

1. **State the problem:** Solve the equation $$\frac{2x+4}{3} = 6$$ for $x$. 2. **Formula and rules:** To solve for $x$ in a fraction equation, multiply both sides by the denominat

1. نبدأ بكتابة العبارة المعطاة: $$x - p\left(-2(x^2 + 1)\sin(2x) - \frac{x \cos(2x)}{(x^2 + 1)^{3/2}}\right)$$

1. **Problem Statement:** Given the quadratic equation $px^2 + qx + 9 = 0$ with roots $\alpha$ and $\beta$, prove that $$\frac{\sqrt{p} + \sqrt{q}}{\sqrt{p} - \sqrt{q}} = 0.$$ 2. *

1. **State the problem:** Find the standard and general form of the line passing through the points $(-10,1)$ and $(-7,2)$. 2. **Formula for slope:** The slope $m$ of a line throug

1. Problem 1: Given the arithmetic sequence $(2m-5, 4, 9)$, find $m$ and determine if the sequence is increasing or decreasing. 2. Recall that in an arithmetic sequence, the differ