🧮 algebra

1. **State the problem:** Simplify the expression $$\log(2)\sqrt{2x+2\sqrt{x^2-1}} + \log(2)(\sqrt{x+1} - \sqrt{x-1})$$. 2. **Recall important formulas and rules:**

1. **State the problem:** We are given the area and length of a rectangle and need to find an expression for its width. 2. **Recall the formula for the area of a rectangle:**

1. **State the problem:** Simplify the expression $\log 2\sqrt{2x+2\sqrt{x^2-1}} + \log 2(\sqrt{x+1} - \sqrt{x-1})$. 2. **Recall logarithm properties:**

1. **Stating the problem:** Simplify the expression $\log_2\left(\sqrt{x+1} - \sqrt{x-1}\right)$. 2. **Recall the logarithm and radical properties:** The logarithm base 2 is define

1. **Problem statement:** A motorcyclist and a cyclist start from points A and B, 180 km apart, and ride towards each other. They meet after 2 o'clock and continue without stopping

1. The problem is to find the equation of the line passing through the points (0,6) and (3,0). 2. The general form of a line is given by the equation $y = mx + c$, where $m$ is the

1. **State the problem:** Simplify the expression $-3m^2 + 3m^2 - m + 2m$. 2. **Identify like terms:** Like terms are terms that have the same variable raised to the same power. He

1. The problem is to simplify the expression or mathematical statement you have. 2. Since you did not provide a specific expression, I will explain the general approach to simplifi

1. **State the problem:** Simplify the expression $$\frac{4}{b} - \frac{1}{a}$$. 2. **Recall the formula for subtraction of fractions:** To subtract fractions, find a common denomi

1. **State the problem:** We need to find how many games United City lost given the total wins, draws, losses, and games played by three teams. 2. **Given data:**

1. **State the problem:** Solve the equation $$2^{x+1} + 2^x = 3$$ for $x$. 2. **Recall the properties of exponents:**

1. **State the problem:** Calculate $3.7 \times 16.2^2 - 500$ and express the answer in different forms. 2. **Calculate the square:** First, find $16.2^2$.

1. The problem is to expand the expression $a^2 + b^2 - 3ab(a + b)$.\n2. First, distribute $-3ab$ over $(a + b)$ using the distributive property: $-3ab \times a = -3a^2b$ and $-3ab

1. The problem is to expand the expression $a^3 + b^3$. 2. We use the sum of cubes formula: $$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$.

1. The problem is to compose a mathematical expression or equation without providing the answer. 2. In algebra, composing an expression means writing it down using variables, const

1. The problem is to understand the rule for expanding $a^4 + b^4$. 2. The expression $a^4 + b^4$ is a sum of fourth powers, which does not factor using the simple rule of adding o

1. **Problem:** Solve by factorization the quadratic equation $$x^2 + 5x + 4 = 0$$. 2. **Formula and rules:** To solve a quadratic equation by factorization, we express it as a pro

1. Problem statement: We want to understand how the term $3ab$ appears in the expansion of $(a+b)^3$. 2. Recall the binomial expansion formula for cubes:

1. **Problem Statement:** Sketch the graph of the function \(y = \frac{6}{x+1}\) using transformations. Identify the domain, range, intercepts, and asymptotes.

1. Problem statement: We are given $a^3 + b^3 = 217$ and $a + b = 7$. 2. Goal: Find $ab$.

1. The problem is to find the easiest way to compute the factorial of a number $n$, denoted as $n!$. 2. The factorial of a non-negative integer $n$ is defined as the product of all