🧮 algebra

1. **Problemstellung:** Wir sollen für welche ganzen Zahlen $x, y$ gilt: $$(9 - 4\sqrt{2})^2 = x + y\sqrt{2}.$$\n\n2. **Formel und Regeln:** Um den Ausdruck zu quadrieren, verwende

1. **Stating the problem:** Simplify the expression $$60\sqrt{4^{10}} \cdot \sqrt[6]{4^{18}} : 4^2 = x\sqrt{4^y}$$ and find $x$ and $y$. 2. **Recall the rules:**

1. **State the problem:** Calculate the value of the expression $$ (200\% \times 0.2) + (172\% \times 0.1) + (169\% \times 0.15) + (200\% \times 0.3) + (0\% \times 0.25) $$ and exp

1. مسئله: معادله را بنویسیم. 2. معادله به صورت کلی می‌تواند شامل متغیرها و ضرایب باشد، مانند:

1. **State the problem:** Solve the equation $e^{x^2} - 2x = 1$ for $x$. 2. **Rewrite the equation:**

1. مسئله را بیان می‌کنیم: الف) تابع $y = (x^2 + 1)^3 (5x - 1)$ را بررسی می‌کنیم.

1. The problem asks to find the decay factor given a population decrease of 3.2%. 2. The decay factor is calculated using the formula:

1. **State the problem:** We need to find the decay factor for an account that decreases by 1.8% each year. 2. **Formula and explanation:** The decay factor is calculated as $1 - r

1. **State the problem:** You invested 4000 in an account that decreases each year. After 1 year, the amount is 3920, and after 2 years, it is 3841.60. We need to find the decay fa

1. The problem states that the bird population is decreasing by 2% per year. 2. The decay factor represents the multiplier applied each year to the population to account for the de

1. **State the problem:** A doctor prescribes 550 mg of a drug, and after 4 hours, 330 mg remains. We need to find the decay factor. 2. **Formula used:** The amount of drug remaini

1. **State the problem:** Find the vertex of the parabola given by the function $$f(x) = -x^2 - 12x - 34$$. 2. **Recall the vertex formula:** For a quadratic function $$f(x) = ax^2

1. **State the problem:** Determine the domain and range of the parabola given by $$f(x) = 3x^2 - 24x + 47$$. 2. **Recall the domain of a quadratic function:** The domain of any qu

1. **State the problem:** Rewrite the linear equation $$20x + 13y = 13$$ in slope-intercept form, which is $$y = mx + b$$ where $$m$$ is the slope and $$b$$ is the y-intercept. 2.

1. **State the problem:** Rewrite the equation $5x - 5y = -17$ in slope-intercept form, which is $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept. 2. **Start with the

1. **State the problem:** We know that 28,700 employees represent 19.3% of the total number of employees in the company. We need to find the total number of employees. 2. **Formula

1. **State the problem:** Solve the equation $x - 11 = \sqrt{41 - 4x}$ for $x$. 2. **Formula and rules:** To solve equations involving square roots, isolate the square root term an

1. **State the problem:** Solve the equation $$\sqrt{3n + 11} = \sqrt{n + 5}$$ and check for extraneous solutions. 2. **Formula and rules:** To solve equations involving square roo

1. El problema es simplificar la expresión $$\sqrt[3]{-54}$$. 2. Recordemos que la raíz cúbica de un número negativo es el negativo de la raíz cúbica del valor absoluto de ese núme

1. **State the problem:** Evaluate the expression $$10 \log\left(\frac{10^{78}}{10} + \frac{3 \times 10^{64}}{10} + \frac{6 \times 10^{54}}{10}\right)$$ to 2 significant figures. 2

1. **State the problem:** Simplify the expression $$7^1 y^3 y^3 y^3$$. 2. **Recall the rules:** When multiplying powers with the same base, add the exponents: $$a^m \times a^n = a^