🧮 algebra

1. The problem is to calculate the value of $r_1^*$ given by the expression $r_1^* = 4.640 \times 0.083 + 0.0000020 \times 5,585,000 \times \sqrt{0.083}$. 2. First, multiply $4.640

1. مسئله را بیان می‌کنیم: مجموع ریشه‌های معادله $$8 - x = \sqrt{x} - 3x - 9\sqrt{x} - x^9$$ را پیدا کنیم. 2. ابتدا معادله را بازنویسی می‌کنیم:

1. **Stating the problem:** Given the quadratic equation $$2y^2 - 5y - 3 = 0$$, find the roots \(\alpha\) and \(\beta\). 2. **Formula used:** For a quadratic equation $$ay^2 + by +

1. مسئله را بیان می‌کنیم: اگر $x=\alpha$ ریشه معادله $$\frac{4}{x-2} = \frac{\sqrt{x}}{\sqrt{x} + \sqrt{2}}$$ باشد، مقدار $$\sqrt{2\alpha - 4}$$ را بیابید. 2. ابتدا معادله را ساده

1. The problem states that \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(2y^2 - 5y - 3 = 0\). We need to find the value of \(\alpha + \beta\). 2. Recall the su

1. **Problemstellung:** Wir sollen die Gleichungen aus Aufgabe 3 lösen und die Lösungen unter der Waage angeben. 2. **Formeln und Regeln:**

1. The problem states: Given $a - b = \frac{14}{3}$, find $a^2 - b^2$. 2. Recall the algebraic identity: $$a^2 - b^2 = (a - b)(a + b)$$

1. **State the problem:** Solve the equation $\frac{y}{x} = 1.1$ for $y$ in terms of $x$. 2. **Formula and rules:** The equation $\frac{y}{x} = 1.1$ means that $y$ divided by $x$ e

1. **State the problem:** We are given the equation $\frac{y}{x} = 6.6$ and need to express $y$ in terms of $x$. 2. **Formula and rules:** The equation shows a ratio of $y$ to $x$.

1. The problem asks us to simplify the expression $16^{\frac{3}{4}}$. 2. Recall the rule: the $n$-th root of a number $a$ is $a^{\frac{1}{n}}$. Therefore, $a^{\frac{m}{n}} = \left(

1. The problem asks to simplify the expression $16^{\frac{3}{4}}$. 2. Recall the rule: the $n$-th root of a number $a$ is $a^{\frac{1}{n}}$. Therefore, $a^{\frac{m}{n}} = \left(a^{

1. **State the problem:** Simplify the expression $2\sqrt{48} - \frac{6}{\sqrt{3}}$. 2. **Recall the rules:**

1. **State the problem:** We need to find the number of adults at a funfair given relationships between adults, boys, and girls.

1. **Stating the problem:** We are given the quadratic function $$y = -\frac{1}{2}x^2 - x + 1$$ and need to analyze its properties.

1. The problem is to solve the equation $2x + 3 = 11$ for $x$. 2. We use the basic algebraic principle that to isolate $x$, we need to undo the operations applied to it. Here, $x$

1. **State the problem:** We are given the quadratic function $$y = -\frac{1}{2}x^{2} - x + 1$$ and need to analyze its properties. 2. **Formula and rules:** A quadratic function i

1. The problem is to simplify the expression $$\sqrt[4]{256} \cdot 81$$. 2. Recall that the fourth root of a number $a$ is a number $b$ such that $b^4 = a$.

1. **State the problem:** Simplify the cube root of the fraction $\frac{27}{8}$, written as $\sqrt[3]{\frac{27}{8}}$. 2. **Recall the property of cube roots:** The cube root of a f

1. **State the problem:** We have a sequence defined by the recurrence relation $$u_{n+3} = u_n + 2$$ with initial values $$u_1 = 0$$, $$u_2 = 1$$, and $$u_3 = 5$$. We need to find

1. **State the problem:** Simplify the radical expression $$\sqrt[3]{\frac{81x^{2}y^{8}}{3x^{8}y^{2}}}$$ and find which option is equivalent. 2. **Write the expression inside the c

1. **State the problem:** We are given the quadratic function $y = -4x^2 + 32x - 57$ and need to analyze it. 2. **Formula and rules:** A quadratic function is generally written as