🧮 algebra

1. **State the problem:** Calculate the expression $$(3a+3b+3c)(27a+27b+27c) \div (9bc+9ca+9ab).$$ 2. **Rewrite the expression:** Factor out common terms:

1. **State the problem:** Calculate the expression $$\frac{(3a+3b+3c)(27a+27b+27c)}{9bc+9ca+9ab}$$. 2. **Rewrite the expression:** Factor out constants from each term:

1. **State the problem:** Ben buys 10 books. The first book costs 3. Each subsequent book doubles in price. After the 6th book, each additional book gets a 15% discount. 2. **Defin

1. **Problem:** Find the equation of the line passing through points $A=(1,3)$ and $B=(-2,-3)$. 2. **Formula:** The slope $m$ of a line through points $(x_1,y_1)$ and $(x_2,y_2)$ i

1. The problem is to evaluate the expression given, which appears to be a long, complex, and unclear sequence of numbers and operations. 2. However, the expression contains many sy

1. **Problem Statement:** Determine if the function $f(x)$ described by the plot is one-to-one and onto. Also, find the domain and range of $f(x)$. 2. **One-to-One Function:** A fu

1. **State the problem:** Find the intersection point of the lines given by the equations $$4x - 18 = -3y$$ and $$5x - 2y = 11$$. 2. **Rewrite the equations in standard form:**

1. **Énoncé du problème :** Justifier graphiquement que, pour tout $x \in ]0,+\infty[$, on a $g(x) - h(x) > 0$ avec $g(x) = x^2$ et $h(x) = 2 \ln x - (\ln x)^2$.

1. **State the problem:** Hashim bought maize for 70000 and spent 20500 to convert it into maize flour. He obtained 3 bags of maize flour, each weighing 25 kg. We need to find the

1. **State the problem:** We are given that a number $N$ decreased by 12% equals 132. We need to find the original number $N$. 2. **Write the expression for 12% decrease:** A 12% d

1. The problem is to find the set of values for $x$ such that $x \in \{1, 2\}$ and $x \geq -1$. 2. The set $\{1, 2\}$ means $x$ can only be 1 or 2.

1. נניח כי יש לנו את הטענה: $$\sum_{k=1}^n \frac{k+2}{k(k+1)2^k} = 1 - \frac{1}{(n+1)2^n}$$

1. **Problem statement:** We need to find a whole number $n$ such that the product $n \times (n + 42)$ is a prime number. 2. **Recall the definition of a prime number:** A prime nu

1. **Stating the problem:** We need to sketch the graph of the equation $$2X - y - 3 = 0$$.

1. **Problem statement:** The first, fifth, and eighth terms of an arithmetic progression (AP) with common difference $3$ form the first three terms of a geometric progression (GP)

1. **State the problem:** We have an arithmetic progression (AP) where the first term $a_1 = 2$.

1. **State the problem:** Solve the complex equation $$p(z) = z^2 - 3z^2 + z + 5$$ for $z$. 2. **Simplify the polynomial:** Combine like terms:

1. **State the problem:** Find the intersection point of the system of equations: $$\begin{cases} 2x + y = 8 \\ 2x + 4y = 20 \end{cases}$$

1. The problem is to analyze and understand the function $$y=2-\sqrt{2-x}$$ and describe its graph. 2. The function involves a square root, so the expression inside the root must b

1. **Problem statement:** Simplify the expression $\sqrt{55 + 30\sqrt{2}}$. 2. **Formula and approach:** We try to express the expression inside the square root in the form $(a + b

1. **State the problem:** We need to find the value of $f(-2 - g(3))$ given the functions $f(x) = \frac{3}{4}x + 10$ and $g(x) = x^2 - 3$. 2. **Calculate $g(3)$:** Substitute $x=3$