🧮 algebra

1. **State the problem:** Simplify the expression $$\frac{2}{1 - \sqrt{3}}$$. 2. **Formula and rule:** To simplify a fraction with a radical in the denominator, multiply numerator

1. The problem states that the sequence starts at 12 and each term increases by 6. 2. This is an arithmetic sequence where the first term $a_1 = 12$ and the common difference $d =

1. The problem is to find the first term in a Fibonacci-type sequence where the second term is 5 and the fifth term is 23. 2. Recall the Fibonacci-type sequence rule: each term is

1. The problem is to simplify the expression $2^{3} \times 4 \times 2^{-1}$. 2. Recall the rules of exponents: when multiplying powers with the same base, add the exponents: $a^{m}

1. **State the problem:** Solve the equation $16^x = 8^{2x+1}$ for $x$. 2. **Recall the formula and rules:** We can express both sides with the same base to compare exponents. Note

1. The problem is to solve the equation $\frac{1}{4} = 8 \times x$ for $x$. 2. The equation states that one quarter equals eight times some number $x$.

1. The problem is to find the sum of the repeating decimals $0.1\dot{}$ and $0.2\dot{8}$.\n\n2. First, let's express each repeating decimal as a fraction.\n\n3. For $0.1\dot{}$, th

1. **بيان المسألة:** لدينا المتتالية \( (u_n) \) المعرفة ب \( u_0 = 3 \) و \( u_{n+1} = 3u_n - 2 \). المطلوب حساب الحدود \( u_1, u_2, u_3 \). 2. **حساب الحدود:**

1. **State the problem:** Simplify the function $$f(x) = \frac{\sqrt{x-6} + \sqrt{2x-2}}{\sqrt{x-1} - \sqrt{x-5}}.$$\n\n2. **Identify the domain:** For the square roots to be defin

1. The problem is to find the domain of a function, which means determining all possible input values ($x$) for which the function is defined. 2. The domain depends on the type of

1. **State the problem:** Simplify the function $$f(x) = \frac{\sqrt{x-6} + \sqrt{2x-2}}{\sqrt{x-1} - \sqrt{x-5}}.$$\n\n2. **Identify domain restrictions:** The expressions under t

1. **State the problem:** We are given the quadratic function $$f(x) = -2x^2 + 5x + 3$$ and asked to complete the table for the domain $$-2 \leq x \leq 5$$, graph the function incl

1. **Problem statement:** We have a parabola $y=f(x)$ opening downwards passing through the origin and peaking near $(0.5,7)$. A point $P(-2,11)$ lies outside the parabola. We want

1. **State the problem:** Find the product of the radicals in the expression $3\sqrt{2}(5\sqrt{12} - \sqrt{18} + 4\sqrt{24})$. 2. **Recall the formula and rules:**

1. Problem. Solve the quadratic equation $2x^2 - 3x - 5 = 0$.

1. **State the problem:** Find the point $(a,b)$ on the line $2x - 3y + 6 = 0$ that is closest to the point $(3,1)$, then find $a+b$. 2. **Formula and concept:** The closest point

1. The problem asks to find the values of $b$, $c$, and $d$ in the cubic function $y = x^3 + bx^2 + cx + d$ given its graph, and to find the coordinates where the curve crosses the

1. **Stating the problem:** Simplify and evaluate the expression $$12 - (-3) \times \left\{ \frac{1}{4} \div \left[-\frac{1}{2}\right]^2 \right\}$$. 2. **Recall the order of operat

1. **Problema:** Determinare dominio, segno, limiti agli estremi del dominio e asintoti della funzione $$f(x) = \frac{1 - x^3}{4x^2 - 1}$$. 2. **Dominio:** Il denominatore non può

1. **State the problem:** We are given two functions: $$f(x) = \frac{3}{x^2}$$

1. **Problem statement:** Given $a=3.0$, $b=4.24$, and $c=3.654$, find the percentage error of the expression $$\frac{a+b}{c+a}$$. 2. **Formula and explanation:** Percentage error