🧮 algebra

1. **State the problem:** Simplify the expression $$exy \left( \frac{y^2}{x} - \frac{2y}{x^2} + \frac{2}{x^3} \right)$$ and understand the next steps. 2. **Distribute the terms:**

1. **State the problem:** Gemechu bought sheep and sold them for 4650 birr. He made a 7% profit on the selling price. We need to find how much he originally paid for the sheep.

1. We are asked to factorize the expression $$(2x + 9)^2 - (x + 4)^2$$. 2. Recognize this as a difference of squares, which has the formula $$a^2 - b^2 = (a - b)(a + b)$$.

1. Let's start with the equation given: $$2x^2 + 2x^2 = (ab)^2$$

1. Stated problem: Simplify the expression $2x^2 + 2x^2$. 2. Identify like terms: Both terms contain $x^2$ and can be added together.

1. Énonçons le problème : factoriser l'expression $ (2x + 5)(6x + 4) $. 2. L'expression est un produit de deux binômes. Pour factoriser, on peut développer et simplifier, puis cher

1. Express each of the following in the form $a - bi$.\\ a) Simplify $\frac{i}{3 - i}$. Multiply numerator and denominator by the conjugate $3 + i$ to get:\\

1. **State the problem:** We need to show the value of the expression $4^5 \cdot 64^3 \cdot 2^3$. 2. **Rewrite bases in terms of prime factors:**

1. **State the problem:** We have 40 questions. \nFraction of multiple choice questions is $\frac{2}{5}$. \nFraction of true or false questions is $\frac{1}{4}$. \nWe need to find

### Exercice 1: Finding limits of sequences 1. **Problem:** Find the limit of \( u_n = \frac{\sin(n)}{n^{2} + 3} \).

1. **State the problem**: Isaac approximated $20 \times 300 = 6000$ by rounding each number to 1 significant figure. We need to find the smallest possible exact product of the orig

1. **State the problem:** There are vans, lorries, and motorbikes in the ratio 2 : 5 : 9. 2. We know that the number of vans corresponds to 2 parts in the ratio.

1. Stating the problem: Solve the system of equations: $$x + y = 5$$

1. Problem: Find the slope of the line through points (1, 3) and (4, 9). Step 1: Use the slope formula $$m = \frac{y_2 - y_1}{x_2 - x_1}$$.

1. Stating the problem: We are given the matrix $$\begin{pmatrix} a - b - c & 2a & 2a \\ 2b & b - c - a & 2b \\ 2c & 2c & c - a - b \end{pmatrix}$$

1. The question asks whether a groupoid and a binary operation are the same and if they both come under algebraic structures. 2. A \textbf{binary operation} on a set $S$ is a funct

1. We are given a cubic equation $$x^3 + ax^2 + bx + c = 0$$ with roots $$\alpha, \beta, \gamma$$ and the system: $$\alpha u + \beta v + \gamma w = 0,$$

1. **Problem statement:** We need to find the values of $a$ and $b$ such that the coefficients of $x^3$ and $x^4$ in the expansion of $\left(1 + ax + bx^2\right)(1 - 2x)^{18}$ are

1. The problem is to simplify the expression **-2(3m - n + 4)**. 2. Distribute the **-2** to each term inside the parentheses:

1. **State the problem:** Find the value of the sum $$\sum_{r=1}^n {^nC_r} \alpha_r$$ where

1. We are given four pairs of simultaneous equations to solve for $x$ and $y$: (1) $x + y + xy = 5$, $x^2 + y^2 = 1$