🧮 algebra

1. **Problem Statement:** Simplify the expression $$\frac{\left[4^{5} \cdot (64)^{3} \cdot 2^{3}\right]^{\frac{1}{2}}}{8^{5} \cdot (128)^{2}}$$

1. Let's start by stating the problem: A mixed inequality involves two inequalities joined together, such as $a < x \leq b$. It means $x$ is a number that satisfies both inequaliti

1. **State the problem:** Solve the equation $\frac{1}{2}(4x+8)=10$ for $x$. 2. **Use the distributive property:** Multiply both sides by 2 to eliminate the fraction:

1. **State the problem:** Solve the equation $9 - x = 3x + 1$ for $x$. 2. **Write down the equation:**

1. The problem asks to write the given numbers as non-terminating decimal fractions (كسور عشرية غير زائدة). 2. To convert a number to a decimal fraction, we express it as a fractio

1. **State the problem:** Simplify the expression $\sqrt{10} + 4\sqrt{10}$. 2. **Recall the rule:** When adding terms with square roots, you can only combine them if the radicands

1. **Énoncé du problème :** Soit la suite $(U_n)$ définie par $U_0 = 1$ et $U_{n+1} = 5U_n - 3$ pour tout $n \geq 0$. Calculer $U_1$, $U_2$ et $U_3$.

1. **State the problem:** Simplify the expression $$\frac{4}{\sqrt[4]{5}+1}$$. 2. **Formula and rule:** To simplify expressions with radicals in the denominator, we rationalize the

1. Задача: Упростить выражение $$\frac{10}{\sqrt{19}+3}$$ и избавиться от иррациональности в знаменателе. 2. Формула и правило: Чтобы избавиться от иррациональности в знаменателе,

1. The problem is to solve the equation or expression given on the full page. Since no specific equation or expression is provided, I cannot solve it directly. 2. To solve any alge

1. **Problem:** Simplify the expression $(3 - 4i) + (6 + i)$. 2. **Formula and rules:** To add complex numbers, add their real parts and their imaginary parts separately: $ (a + bi

1. **Problem Statement:** A quantity increases by 200% over 60 units of time. At the end of each year, it becomes 2.5 times the previous amount. We want to find after how many year

1. Задача: Упростить выражение $$\frac{11}{\sqrt{17}-\sqrt{6}}$$, избавившись от иррациональности в знаменателе. 2. Формула: Чтобы избавиться от иррациональности в знаменателе, умн

1. **State the problem:** Simplify the expression $\frac{2}{\sqrt[5]{2}}$. 2. **Recall the rule:** The $n$th root of a number can be written as a fractional exponent: $\sqrt[n]{a}

1. The problem is to evaluate the expression $3 \times 3 - 3 \div 3 + 3$. 2. According to the order of operations (PEMDAS/BODMAS), multiplication and division are performed before

1. **State the problem:** Simplify the expression $\frac{1}{\sqrt[3]{4}}$. 2. **Recall the formula:** The cube root of a number $a$ is written as $\sqrt[3]{a} = a^{\frac{1}{3}}$.

1. **Problem:** Find the domain and range of the function $$f(x) = \sqrt{2x - 3}$$. 2. **Domain:** The expression inside the square root must be non-negative because the square roo

1. The problem is to find the center (midline) of the sinusoidal function $f(x)$ which oscillates between $-2.0$ and $2.0$ on the y-axis. 2. The center or midline of a sinusoidal f

1. Задача: Упростить выражение $$\frac{8}{\sqrt[4]{5}}$$, избавившись от иррациональности в знаменателе. 2. Формула и правило: Чтобы избавиться от корня в знаменателе, нужно умножи

1. **بيان المسألة:** لدينا أعداد حقيقية $x$ و $y$ بحيث $7 \leq x \leq 5$ و $8 \leq y \leq -4$. نريد إيجاد حصر الأعداد التالية: $x + y$, $x - y$, $y^2 - x^2$, $\frac{y}{x}$، و $\fra

1. **Problem statement:** A train starts its journey from station P at 7:00 am, reaches station Q at 7:45 am, and station R at 8:25 am. The mean speed of the train is 100 km/h. The