🧮 algebra

1. مسئله: عبارت $$\frac{7}{5}(\sqrt{9} - \sqrt{6}) - \frac{1}{\sqrt{7} - \sqrt{6}}$$ را ساده کنید. 2. ابتدا مقادیر رادیکال‌ها را محاسبه می‌کنیم:

1. مسئله: گویا کردن مخرج کسر یعنی حذف عبارت‌های گویا (کسرهای داخل مخرج) به طوری که مخرج به صورت یک چندجمله‌ای ساده شود. 2. فرمول و قانون مهم: برای گویا کردن مخرج کسرهایی که شامل جم

1. **Énoncé du problème :** Trouver la valeur de $c$ dans la fonction $f(x) = a c^x + 5$ en utilisant les équations données. 2. **Données fournies :**

1. The problem asks to find the value of $p^2 + q^2$ given $p = 5 + 2\sqrt{6}$. However, the value of $q$ is not provided in the problem statement, so we cannot directly compute $p

1. **Problem:** Given $p = 5 + 2\sqrt{6}$, find the value of $p^2 + p - 1$. 2. **Formula and rules:** To find $p^2 + p - 1$, first calculate $p^2$, then add $p$, and subtract 1.

1. **State the problem:** Simplify the expression $$\frac{7}{3} - \frac{1}{3} \times \frac{7}{2} + \frac{15}{8} \times \frac{4}{5} - 3 \div \left( \frac{3}{4} \times \frac{3}{2} \r

1. **Stating the problem:** We are given the complex number $11 + 2i$ and asked to express it in the form $d + ie$ where $d$ and $e$ are real numbers. 2. **Understanding complex nu

1. Evaluate each power without using a calculator. **a)** $16^{1/2}$ means the square root of 16.

1. **State the problem:** Find the equation of the line passing through points C(10, 15) and D(-8, -21) in the form $y = mx + c$.

1. Problem: Determine for which values of the parameter $k$ the function $f$ is defined for every real number $x \in \mathbb{R}$. We analyze each function separately, focusing on t

1. The problem is to simplify and analyze the function $$y=\frac{\sqrt{x}-1}{x+1}$$. 2. The formula involves a square root in the numerator and a linear expression in the denominat

1. **Stating the problem:** We have a rectangular prism with side lengths $x$, $x+2$, and $x+3$. The volume is given as 90 cm³. 2. **Formula used:** The volume $V$ of a rectangular

1. **Énoncé du problème :** Soit $x \in \mathbb{R}$ tel que $-2 < x < 0$, on pose $A = x^2 + x - 2$ et $B = \frac{x}{x-1}$.

1. **State the problem:** Convert the fractions $\frac{2}{5}$, $\frac{7}{12}$, $\frac{3}{4}$, and $\frac{-5}{6}$ to equivalent fractions with a common denominator. 2. **Find the Le

1. **State the problem:** We need to convert the fractions $\frac{3}{4}$, $\frac{9}{22}$, $\frac{7}{8}$, and $-\frac{5}{11}$ to equivalent fractions with a common denominator calle

1. **State the problem:** Solve the equation $-4(p + 2) + 8 = 2(p - 1) - 7p + 15$ for $p$. 2. **Apply the distributive property:**

1. **State the problem:** We need to graph the line given by the equation $$y + 6 = -\frac{1}{2}(x + 10)$$. 2. **Rewrite the equation in slope-intercept form:** The slope-intercept

1. مسئله: عدد رادیکالی $\sqrt{18}$ را به صورت عدد گویا بین دو عدد 3 و 8 بنویسیم. 2. ابتدا رادیکال را ساده می‌کنیم. چون $18 = 9 \times 2$ و $9$ یک مربع کامل است، داریم:

1. **State the problem:** Find the equation of the line passing through the points $(1,-5)$ and $(9,11)$. 2. **Formula used:** The equation of a line in slope-intercept form is $y

1. **State the problem:** Solve the system of linear equations: $$\begin{cases} 2x_1 - 5x_2 - 6x_3 = 5 \\ -3x_1 + 2x_2 + 5x_3 = 1 \\ 2x_1 + 4x_2 - 3x_3 = -16 \end{cases}$$

1. The user message lists multiple lessons and skills related to algebra topics such as solving systems of equations, graphing quadratic functions, complex numbers, and polynomial