🧮 algebra

1. **State the problem:** Simplify each expression involving division of powers with the same base. 2. **Formula used:** For any base $a \neq 0$ and integers $m, n$, the rule is:

1. The problem is to find the y-coordinate of the vertex of a parabola given by a quadratic function. 2. The general form of a quadratic function is $$y = ax^2 + bx + c$$ where $a$

1. Write as a fraction: a) $a t^{-1} = \frac{a}{t}$

1. **Problem statement:** Given functions $f(x) = 3x + 2$, $g(x) = x^2 + 1$, and $h(x) = 4^x$, solve the following: (a) Find $h(3)$.

1. Simplify expressions with zero exponents: - Any nonzero number raised to the zero power equals 1: $a^0 = 1$ if $a \neq 0$.

1. **Énoncé du problème :** Développer puis réduire les expressions suivantes : - $-2a(3a-4)$

1. **State the problem:** Simplify the expression $$\frac{4R^2 - 2a^2}{\sqrt{4R^2 - a^2}}$$. 2. **Identify the components:** The numerator is $$4R^2 - 2a^2$$ and the denominator is

1. **Problem Statement:** Solve the equation $$|x - 3|^{3x^2 - 10x + 3} = 1$$ and explain why $$x = 3$$ is rejected but $$x = \frac{1}{3}$$ is accepted. 2. **Formula and Rules:** F

1. Problem 4a: Write a system of linear inequalities for the theater ticket sales. Let $x$ be the number of adult tickets and $y$ be the number of youth tickets.

1. **Problem Statement:** Given two functions $f(x) = 7x + 1$ and $g(x) = 5x + 8$, find the composite functions $(f \circ g)(x)$ and $(g \circ f)(x)$. 2. **Formula and Explanation:

1. **Problem 1:** Complete the sequence where each term is of the form $\text{number} \times 8 + \text{digit}$. 2. **Observation:** The pattern shows that multiplying the number fo

1. Problem 17: Given the operation $a \otimes b = \frac{a^2}{b}$ for all nonzero numbers, find $$[(1 \otimes 2) \otimes 3] - [1 \otimes (2 \otimes 3)].$$ 2. Use the definition of t

1. **تحليل العبارات الجبرية:** - العبارة الأولى: $E=5a-10$

1. Let's clarify the problem: you mentioned "the equation above," but no specific equation was provided in this message. 2. Please provide the exact equation or problem statement y

1. Let's clarify the problem: You mentioned that the equation is "the other way round," meaning the denominator becomes the numerator and vice versa. 2. Suppose the original equati

1. **State the problem:** Given the equation $\frac{x}{y} = 2.5$, find the values of $x$ and $y$. 2. **Understand the equation:** The equation $\frac{x}{y} = 2.5$ means that $x$ is

1. **Problem Statement:** We are exploring the discriminant (Δ) of a quadratic equation and its implications on the roots and graph of the equation. 2. **Definition and Formula:**

1. **Problem Statement:** Solve each system of linear equations using the matrix method, i.e., express as $AX = B$ and find $X = A^{-1}B$ if $A$ is invertible. 2. **Matrix Method F

1. **Problem statement:** (a) A squash club has 27 members. 19 have black hair, 14 have brown eyes, and some have both black hair and brown eyes.

1. **Problem:** Aunt Anna is 42 years old. Caitlin is 5 years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin? 2. **Problem:** Which of these numb

1. **State the problem:** A gardening service charges a base fee of 45 plus 16 for each hour of work. The total cost is 253. We need to find the number of hours $h$ worked. 2. **Wr