🧮 algebra

1. **State the problems:** Solve the quadratic equation $9x^2 - 3x + 1 = 0$ and then prove that $$81\left(x^4 + \frac{1}{6561x^4}\right) = 1.$$

1. **Problem:** Calculate the time Mofenvl took to travel from his cattle post at 23:10 to his village at 01:08 the following day. 2. We notice the arrival time is past midnight, s

1. لنفترض أن الدالة الأصلية هي دالة $y=f(x)$، حيث يجب عليك تحديد شكل الدالة الأصلية لتتمكن من رسم الدالة العكسية. 2. لحساب الدالة العكسية $f^{-1}(x)$، يجب حل المعادلة $y=f(x)$ بالن

1. **Problem statement:** We need to graph two functions using the point plotting method for integer values of $x$ from $-3$ to $3$.

1. **المشكلة الأولى:** لدينا التابع $f: \mathbb{R}^+ \to \mathbb{R}^+, f(x) = (x + 3)^2$. المطلوب هو: 1.1 أوجد التابع العكسي $f^{-1}(x)$.

1. Solve the equation $5x - 1 = 8x + 1$. Step 1: Subtract $5x$ from both sides:

1. **State the problem:** Solve for $x$ in the equation $$3x + x + x + x - 3 - 2 = 7 + x + x$$. 2. **Combine like terms on the left side:** $$3x + x + x + x = 6x$$ and $$-3 - 2 = -

1. State the problem: Simplify the expression and verify the equation $$\frac{(-4)\times 11\times 23\times \sqrt{26}}{(-38)^{-2}\times 1444} = 2^3 \times 11 \times 23$$. 2. Simplif

1. State the problem: Find the value of $k$ such that $5 - \frac{6}{k} = -13$. 2. Start by isolating the term with $k$:

1. لنبدأ بتحديد نوع المعادلات الرياضية التي تريد حلها. هل هي معادلات خطية، تربيعية، أو معادلات من نوع آخر؟ 2. مثال: حل المعادلة التربيعية $ax^2 + bx + c = 0$، حيث $a\neq 0$.

1. The problem is to simplify the expression $\sqrt{x}32 - \sqrt{x}50$ in terms of surds. 2. First, recognize that $\sqrt{x}32$ and $\sqrt{x}50$ should be written as $32\sqrt{x}$ a

1. The problem is to convert the time durations 30 minutes and 2 hours into a ratio. 2. First, convert all times to the same unit for an accurate ratio comparison. Since 1 hour = 6

1. **State the problem:** We are asked to solve and analyze the function $y = -x + 5$. 2. **Rewrite the equation:** The function is a linear equation, where the slope is $-1$ and t

1. **State the problem:** Simplify the expression $$2 - \sqrt{27} + \frac{(12a^2b^3)^{-1} \times (\sqrt{3} ab)^3}{(4^{-1} a) \times (202b)^0}$$. 2. **Simplify each part step-by-ste

1. Solve the equation $x + \sqrt{x} = \frac{6}{25}$. First, let $y = \sqrt{x}$, then $x = y^2$. Substitute:

1. **Problem statement:** Simplify each of the expressions given. \(\text{(a) } \left(a^8 b^{12}\right)^2 \div \left(a^5 b^7\right)^3 \)

1. The problem is to simplify the expression $\frac{2x}{t} \times \frac{1}{2x}$.\n\n2. Write the expression as a single fraction: $$\frac{2x}{t} \times \frac{1}{2x} = \frac{2x \tim

1. Simplify $\frac{(a^8 b^{12})^2}{(a^5 b^7)^3}$: Apply power rule: $(x^m)^n = x^{mn}$.

1. დავიმახსოვროთ, რომ საერთო გამყოფი (GCD) - არის ორ მთელი რიცხვის ყველაზე დიდი რიცხვი, რომელიც ორივეს ভাগდება მთელიებით. 2. მოცემულია ორი რიცხვა: 120 და 676.

1. დავასახელოთ პრობლემა: უნდა ვიპოვოთ რიცხვების 18 და 24 საერთო გამყოფი. 2. საერთო გამყოფის (საუკეთესო საერთო გამყოფის) მოძებნა: უნდა გავარკვიოთ რიცხვების ძირითადი გამყოფები და ვიპ

1. Let's first write down the given function: $$g(t) = \sqrt{3} - t - \sqrt{2} + t$$