🧮 algebra

1. **State the problem:** Solve the equation $\alpha + 2\beta + 3\lambda = 2$ for one variable in terms of the others. 2. **Formula and rules:** This is a linear equation with thre

1. The problem is to simplify the expression $\frac{3}{3^2}$.\n\n2. Recall the rule for exponents: $a^m \div a^n = a^{m-n}$. Here, the numerator is $3 = 3^1$ and the denominator is

1. **Statement of the problem:** We need to determine if the function $$k(x) = \frac{(x^2)-4}{|x|-1}$$ is even. 2. **Recall the definition of an even function:** A function $$f(x)$

1. **State the problem:** Prove that an irrational number exists. 2. **Explanation:** A number is irrational if it cannot be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$

1. **Stating the problem:** Prove that an even number is divisible by 2. 2. **Definition:** An even number is any integer that can be written as $2k$ where $k$ is an integer.

1. **Nêu bài toán:** Liệt kê các hạng tử của đa thức $A(x) = -4x^3 + 3x^2 - x + 1$. 2. **Định nghĩa hạng tử:** Hạng tử của đa thức là từng đơn thức riêng biệt được cộng hoặc trừ tr

1. Problem: Calculate the value of $l = \left|\left(\frac{8}{3} - 8\right) - \left(-\frac{8}{3} + 8\right)\right|$. 2. First, simplify inside the absolute value step by step.

1. The problem is to understand and analyze the function $y=\frac{1}{3}x^2$. 2. This is a quadratic function in the form $y=ax^2$, where $a=\frac{1}{3}$.

1. **State the problem:** Solve the system of equations: $$\frac{27^x}{81^x} + 2y = 9$$

1. **Problem:** Determine if the function $f(x) = x^4 - 5x^3 + x^2 + 12x - 5$ has a zero between 0 and 1. 2. **Method:** Use the Intermediate Value Theorem which states that if a c

1. **State the problem:** Solve the first system of linear equations for variables $f$ and $g$: $$3(3f + 2g) = 5 - f$$

1. We are given the system of equations: $$\frac{5x}{8} - \frac{y}{2} = \frac{1}{4}$$

1. **Problem:** Find the value of $$\sqrt[4]{81}^{-2}$$. 2. **Formula and rules:**

1. **State the problem:** Solve the equation $\frac{1}{2}(4g - 3) = 5g - 7$ for $g$. 2. **Use the distributive property:** Multiply both sides by 2 to eliminate the fraction.

1. **State the problem:** Mohsin’s earnings increase exponentially at a rate of 8.7% each year. In 2018, he earned 195600. We want to find how much more he earns in 2027 compared t

1. **Problem Statement:** We want to understand the formula of Newton's Binomial Theorem and how it is used to expand expressions of the form $$(a+b)^n$$ where $n$ is a non-negativ

1. Solve the inequality $\frac{2x}{3} - \frac{1}{2} < 0$. Step 1: Add $\frac{1}{2}$ to both sides:

1. **State the problem:** Solve the linear equation $4x + y = 8$ for $y$ in terms of $x$. 2. **Formula and rules:** To isolate $y$, subtract $4x$ from both sides of the equation.

1. **Problem statement:** Given the relation $$R = \{ (x,y) \in \mathbb{R}^2 \mid y^2 = 4x - x^2 \},$$ find the domain and range of $R$, determine if $R$ defines a function from $\

1. **Problem:** Solve the linear inequality $$\frac{2x}{3} - \frac{1}{2} < 0$$. 2. **Formula and rules:** To solve inequalities, isolate the variable on one side. Remember, multipl

1. **State the problem:** Simplify the expression $$(2x - 3)^2 + (2x + 3)^2$$. 2. **Recall the formula:** The square of a binomial is given by $$(a \, \pm \, b)^2 = a^2 \, \pm \, 2