🧮 algebra

1. **State the problem:** Simplify the expression $$[3x+5y+6z]-[2x+3y+3z]-x-y+2z$$. 2. **Apply the distributive property:** Remove the brackets by distributing the minus signs.

1. **State the problem:** Express the quadratic polynomial $2x^2 + 8x + 6$ in complete square form. 2. **Recall the formula:** A quadratic $ax^2 + bx + c$ can be written as $a(x -

1. **State the problem:** We need to solve the equation $$\frac{5 \times 10^{-8} \times 2}{3 \times 20} = 15.8$$ for verification or simplification. 2. **Write the formula and simp

1. The problem is to find the value of the expression $\{-15,3\} + [\sqrt{20}] - \left[\frac{10}{3}\right] + \{8,3\}$. 2. First, clarify the notation: $\{a,b\}$ means the decimal n

1. **State the problem:** We know that $y$ is inversely proportional to $x$, which means $y \propto \frac{1}{x}$. Given $x=5$ when $y=6$, find $x$ when $y=12$. 2. **Formula and exp

1. Énoncé du problème : Calculer et simplifier les divisions de fractions suivantes, puis vérifier avec une calculatrice. 2. Rappel de la règle : Diviser par une fraction revient à

1. **Stating the problem:** We are given the function $$y = c_1 e^{-2x} + c_2 e^{3x}$$ and want to understand its behavior and graph. 2. **Formula and explanation:** This function

1. The problem is to simplify the expression $14b$. 2. Since $14b$ is already in its simplest form, it represents the product of the number 14 and the variable $b$.

1. The problem is to evaluate the expression $-12^{-12}$. 2. Recall the order of operations: exponents are evaluated before negation. So, $-12^{-12}$ means the negative of $12^{-12

1. **State the problem:** Simplify the expression $-12x^{-12}$. 2. **Recall the rule for negative exponents:** For any nonzero number $a$ and integer $n$, $a^{-n} = \frac{1}{a^n}$.

1. **State the problem:** Calculate the value of the expression $[-22,4] + [\sqrt{50}] - \left[\frac{7}{3}\right] + \{1,4\}$. 2. **Interpret the notation:** Assuming $[-22,4]$ mean

1. **State the problem:** We are given a hierarchical tree of mathematical expressions and asked to evaluate the root node labeled "5" based on the expressions and their relationsh

1. **Problem:** Simplify the product $$\left(\frac{5y^3}{32x}\right) \times \left(\frac{-4}{15x^2 y^3}\right)$$ 2. **Step 1: Factor all numerators and denominators completely.**

1. Problem: Multiply the rational expressions \(\frac{5y^3}{32x} \times \frac{-4}{15x^2 y^3}\). 2. Formula: To multiply rational expressions, multiply the numerators and denominato

1. **State the problem:** Decompose the rational expression $$\frac{3x^2 + 4}{(x-3)(x^2 + 2)}$$ into partial fractions. 2. **Formula and rules:** For a rational expression where th

1. **Problem statement:** Factorise each expression completely. 2. **Formula used:** Difference of squares: $$a^2 - b^2 = (a-b)(a+b)$$

1. **Stating the problem:** We have two men and parrots with combined heights given. The first man plus his parrot equals 200 units. The second man plus his parrot equals 170 units

1. **Problem Statement:** We have a rectangular cardboard of dimensions 60 cm by 40 cm. We cut out equal squares of side length $x$ cm from each corner and fold up the sides to for

1. **State the problem:** We have a function $f(x) = (x - 5)(x + a) - bx$ where $a > 0$ and constants $a, b$. When $f(x)$ is divided by $x - 5$, the remainder is 80.

1. Let's start by stating the problem: You want to understand the prime factorization method. 2. Prime factorization is the process of expressing a number as a product of its prime

1. Let's start by stating the problem: We want to find the Least Common Multiple (LCM) of two or more numbers. 2. The LCM of numbers is the smallest positive integer that is divisi