🧮 algebra

1. **State the problem:** Solve the equation $5(3 - 2i) + 2i(4 + 6i) = 0$ for the imaginary unit $i$. 2. **Recall the property of $i$:** $i^2 = -1$.

1. The problem asks to find the positive and negative intervals of a function, which means determining where the function's values are greater than zero (positive) and where they a

1. **State the problem:** We have the function $y=4\sqrt{-8x+16}-11$ and want to find its vertex, domain, and range, and rewrite it in vertex form. 2. **Rewrite the function inside

1. The problem is to simplify an expression, but since no specific expression was given, let's consider a general approach to simplification. 2. Simplification involves reducing an

1. The problem states: Solve for the unknown number $a$ in the equation $\frac{6}{8} = \frac{9}{a}$.\n\n2. We use the property of proportions: if $\frac{x}{y} = \frac{m}{n}$, then

1. The problem is to simplify the given expression (assumed from previous context). 2. To simplify an expression, we combine like terms, apply arithmetic operations, and use algebr

1. **State the problem:** Simplify the expression $\left(s^3 t^5\right)^4$ and write it as a single power. 2. **Recall the power of a product rule:** When raising a product to a po

1. The problem is to find the sum of the series $$\frac{1}{5} + \frac{2}{5} + \frac{3}{5} + \cdots + \frac{100}{5}$$. 2. We can factor out the common denominator 5 from each term:

1. **Problem statement:** Find the inverse function value $f^{-1}(2)$ for the function $$f(x) = \frac{2^x - (\frac{1}{2})^x}{2}.$$\n\n2. **Recall the function and inverse definitio

1. **State the problem:** Verify if the equality \( \frac{5}{12} \times 2 \frac{2}{3} = \frac{5}{15} \times \frac{8^2}{3} = \frac{10}{9} \div \frac{1}{9} \) is correct. 2. **Conver

1. **Problem Statement:** Determine the domain on which the function $y = f(x)$, a downward-opening parabola with vertex near $(-3,4)$, is increasing. 2. **Key Concept:** For a par

1. The problem is to solve the equation or expression given by the user. However, since no specific equation or expression was provided, I cannot solve it directly. 2. To solve an

1. **Stating the problem:** We want to prove or verify the inequality $$3^n < (n+1)!$$ for natural numbers $n \geq 4$. 2. **Recall definitions and formulas:**

1. مسئله: عبارت داده شده را به صورت جبری بنویسید. 2. برای نوشتن عبارت به صورت جبری، باید از نمادهای ریاضی مانند متغیرها، عملگرها و اعداد استفاده کنیم.

1. مسئله: عبارت داده شده را به صورت جبری بنویسید. 2. برای نوشتن عبارت به صورت جبری، باید از نمادهای ریاضی مانند متغیرها (مثلاً $x$, $y$) و عملگرهای جبری (جمع، تفریق، ضرب، تقسیم) اس

1. مسئله: مقدار عبارت $$\left(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ... + \frac{1}{50}\right) + \left(\frac{1}{2} + \frac{2}{3} + \frac{3}{4} + ... + \frac{49}{50}\right) + 2$$

1. Problem: Solve the equation $-9x - 5 = 13$. 2. Formula and rules: To solve linear equations, isolate the variable $x$ by performing inverse operations.

1. **Problem:** Given that $(kx+1)$ is a common factor of the polynomials $2x^2 + 7x + 3$ and $2x^2 - 5x - 3$, find the value of $k$. Then find the remainder when $2x^2 + x^2 - 18x

1. **State the problem:** Solve the equation $1.6x + 1.475(1 - x) = 1.50974$ for $x$. 2. **Write the equation and apply distributive property:**

1. Problemet är att lösa ekvationssystemet för variablerna $x$ och $y$. 2. Antag att ekvationssystemet är av formen:

1. **State the problem:** Solve the equation $$(28 - x) y + \frac{xy}{2} = 600$$ for $y$ in terms of $x$. 2. **Rewrite the equation:** The equation is $$(28 - x) y + \frac{xy}{2} =