🧮 algebra

1. **Problem 10a:** Find a formula for the sum $$\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \cdots + \frac{1}{n(n+1)}$$

1. **State the problem:** Solve the system of equations using the Gauss-Jordan method: $$\begin{cases} 2x + y - z = 8 \\ -3x - y + 2z = -11 \\ -2x + y + 2z = -3 \end{cases}$$

1. **Problem:** Given that $10^x = 64$, find the value of $10^{\frac{x}{2} + 1}$.\n\n2. **Formula and rules:** We use the properties of exponents: $10^{a+b} = 10^a \times 10^b$ and

1. **State the problem:** We have a cubic polynomial $ax^{3}+bx^{2}+cx-4$. 2. When divided by $(x+2)$, the remainder is double the remainder when divided by $(x+1)$.

1. **State the problem:** We need to find the values of $x$ and $y$ such that $$\frac{(1+i)x-2i}{3+i}+\frac{(2-3i)y+i}{3-i}=i.$$\n\n2. **Rewrite the equation:** The equation involv

1. **State the problem:** We need to find the solution to the system of equations \(y = x - 2\) and \(y = -x + 4\) and graph them. 2. **Set the equations equal to find the intersec

1. Diberikan persamaan kuadrat $x^2 - x - 2 = 0$ dengan akar-akar $p$ dan $q$. Kita diminta mencari nilai dari $(p^2 - p - 4)(q^2 - q + 4)$. 2. Pertama, kita gunakan sifat akar-aka

1. **Stating the problem:** Simplify the expression $$x^{4} - x^{2}y^{2} + 16y^{4}$$. 2. **Formula and rules:** This is a polynomial expression involving powers of $x$ and $y$. We

1. نبدأ بكتابة معادلة القطع المكافئ المعطاة: $$\frac{1}{3}x^2 - 4y = 0 \implies y = \frac{1}{12}x^2$$

1. نبدأ بتحديد المشكلة: نريد إيجاد معادلة القطع المخروطي الذي يمس دليل القطع المكافئ. 2. معادلة القطع المكافئ القياسية التي دليلها هو الخط $y = -p$ (حيث $p$ هو البعد البؤري) هي:

1. مسئله: باید مقدار عبارت $$\frac{4^{-15} \times 32^3}{(4^{-5})^5 \times 16^4}$$ را به صورت توان دار ساده کنیم. 2. فرمول‌ها و قوانین مهم:

1. مسئله: مقدار عبارت $$\frac{8^{-1} \times 4^{3}}{2^{-4} \times \left(\frac{1}{8}\right)^{2}}$$ را به صورت عدد توان دار بیابید. 2. فرمول‌ها و قوانین مهم:

1. **Stating the problem:** We want to estimate how much electricity the clinic should purchase for a month based on the given usage data and top-ups. 2. **Given data:**

1. مسئله را بیان می‌کنیم: محاسبه عبارت $$\frac{36^2 \times 25^4 \times 49^2}{6^{-3} \times 5 \times 7^{-3}}$$ به صورت توان‌دار.

1. مسئله: مقدار عبارت $$\frac{3^{-2} \times 2^{-16}}{12^{-5}}$$ را به صورت عدد توان دار بیابید. 2. فرمول‌ها و قوانین مهم:

1. **مسئله:** مقدار عبارت $$\frac{3^{-2} \times 2^{-16}}{12^{-5}}$$ را بیابید. 2. **فرمول‌ها و قوانین مهم:**

1. The problem is to find all solutions to the equation or system given (please specify the exact equation or system for precise solving). 2. Since the user did not provide a speci

1. **Problem:** Sketch the graph of the linear equation $$4x - 2y = -12$$ in slope-intercept form. 2. **Convert to slope-intercept form:** The slope-intercept form is $$y = mx + b$

1. مسئله: محاسبه مقدار عبارت $$\left(\frac{25}{100}\right)^{-7} \times 8^{5}$$. 2. فرمول‌ها و قوانین مهم:

1. The problem is to find the solution region for the system of inequalities: $$y \leq 2x + 1$$

1. The problem is to evaluate the expression $\left(\frac{0}{25}\right)^{-7} \times 8^{5}$. 2. Recall the rule for negative exponents: $a^{-n} = \frac{1}{a^n}$, but this only appli