🧮 algebra

1. **Problem:** Calculate $$\left[\left(\frac{2}{3}\right)^{-4}\right]^{\frac{2}{3}}$$ and $$-1.5 : \left(\frac{3}{2}\right)^{\frac{1}{9}} \cdot 121^{\frac{5}{8}}$$. **Step 1:** Si

1. **Problem statement:** Expand and reduce the polynomial $$Q(x) = (4x + 5)^2 - (x - 2)(4x + 5)$$. 2. **Recall formulas:**

1. The problem asks: For the function $f(x) = -5$, where is it negative? 2. Since $f(x) = -5$ is a constant function always equal to $-5$, it is negative for all real numbers.

1. The problem is to solve the inequality involving $x$ where the solution is given as $x<2$ or $x>3$. 2. This type of inequality typically arises from expressions like $(x-2)(x-3)

1. **Stating the problem:** We want to decompose the function $$y(p) = \frac{1}{p^2 + 4} + \frac{1 - e^{-p}}{p(p^2 + 4)}$$ into partial fractions. 2. **Rewrite the expression:** Co

1. **Problem Statement:** Verify if the functions $f(x) = \frac{1}{2}x - 7$ and $g(x) = 2x - 14$ are inverse functions by using composition of functions. 2. **Formula and Rule:** T

1. Let's start by stating the problem: A quadratic inequality involves an expression of the form $ax^2 + bx + c$ where $a \neq 0$, and we want to find the values of $x$ that make t

1. The problem asks for the "primary capacity" of the complex number $\cos(50) + i \sin(50) + i$. 2. First, recognize that $\cos(50) + i \sin(50)$ is in the form of Euler's formula

1. נניח את הבעיה: נתונה הפונקציה $$h(x) = \frac{x^2 - 1}{x^2 + 1}$$ על התחום $$(-\infty, \infty)$$. 2. נבדוק את תחום ההגדרה: המונה הוא $$x^2 - 1$$ והמכנה הוא $$x^2 + 1$$. מאחר ש-$$

1. ננתח כל טענה בנפרד כדי לבדוק אם היא נכונה או לא. 2. טענה ראשונה: אם $x \notin \mathbb{Q}$ ו-$y \in \mathbb{Q}$ אז בהכרח $x + y \notin \mathbb{Q}$.

1. **State the problem:** We need to divide the polynomial $$x^4 - 8x^3 + (5a - 1)x^2 + 6x - 3a - 6$$ by $$x^2 - 1$$.

1. **Stating the problem:** Multiply the polynomials $$ (x^4 - 8x^3 + (5a - 1)x^2 + 8x - 3a - 6) \cdot (x^2 - 1) $$.

1. **Problem:** Calculate the capacity of a tank that is 50 dm long, 3 dm wide, and 20 dm deep, when it is \(\frac{3}{5}\) full. 2. **Formula:** Volume of a rectangular tank is giv

1. The problem is to find the missing numerator $x$ in the equation $$8 \times \frac{x}{6} = \frac{20}{3}$$. 2. The formula used here is to solve for $x$ in a proportion or equatio

1. مسئله: مقدار عبارت $\frac{\sqrt{1}}{\sqrt{3}}$ را بیابید. 2. فرمول‌ها و قوانین مهم:

1. **Problem statement:** We have a cubic polynomial $p(x)$ such that for $j=1,2,3,4$, $p(\frac{1}{j})=2j-1$. We need to find $p(-1)$. 2. **Understanding the problem:** Since $p(x)

1. **State the problem:** We have a cubic equation $$x^3 + 3x^2 - 24x + 1 = 0$$ with roots $$a, b, c$$. We want to find the value of

1. **Stating the problem:** We analyze the function $f_1: y = x - 3$. 2. **Domain and range:** The function $y = x - 3$ is a linear function defined for all real numbers, so the do

1. **State the problem:** Solve the equation $x + 3 = 2 \left( \frac{1}{3} x - 1 \right) + 4^3$. 2. **Recall the order of operations and distributive property:** First, calculate p

1. **State the problem:** Solve the exponential equation $$9^x - 10 \cdot 3^x + 9 = 0$$. 2. **Rewrite the bases:** Note that $$9 = 3^2$$, so we can write $$9^x = (3^2)^x = 3^{2x}$$

1. Let's start by stating the problem: We want to find the values of $x$ such that $f(x) = 0$. 2. The equation $f(x) = 0$ means we are looking for the roots or zeros of the functio