🧮 algebra

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: $$g(t) = \sqrt{3} - t - \sqrt{2} + t$$

1. **State the problem:** We have the function $$f(x) = \frac{x + 4}{x^2 - 9}$$ and we want to analyze its components and behavior. 2. **Factor the denominator:** The denominator i

1. **State the problem:** Given the function $f(x) = 3x^2 - x + 2$, we need to find the values of $f(2)$, $f(-2)$, $f(a)$, $f(-a)$, $f(a+1)$, $2f(a)$, $f(2a)$, $f(a^2)$, $[f(a)]^2$

1. **Problem statement:** Given a graph of a function $f$, answer the following: (a) Find $f(1)$.

1. Stating the problem: Simplify the expression $$\frac{6}{4} - \frac{3}{5} + \left( \frac{4}{2} \cdot \frac{3}{7} y \sqrt{3ab} \right) 3 - 9 - 2 \sqrt{3}$$

1. **State the problem:** Analyze the rational function $$f(x) = \frac{x + 4}{x^2 - 9}$$ by identifying its domain, vertical asymptotes, horizontal asymptote, and intercepts. 2. **

1. Given the function $f(x) = 3x^2 - x + 2$, we need to find the values of $f(2)$, $f(-2)$, $f(a)$, $f(-a)$, $f(a+1)$, $2f(a)$, $f(2a)$, $f(a^2)$, $[f(a)]^2$, and $f(a+h)$.\n\n2. C

1. **State the problem:** We are analyzing a function $f$ based on its graph.

1. ننص مشكلة السؤال: نريد إيجاد قيم $k$ بحيث يكون للمعادلة $$-3x^2 + (2k + 1)x - 4k = 0$$ جذور حقيقية. 2. لكي تكون المعادلة التربيعية لها جذور حقيقية، يجب أن يكون المميز $\Delta \g

1. Problem: Evaluate the expression $$\frac{22^2 + 22 - 12}{22 + 8}$$ and find the composite function $f \circ g(3)$ given \(g(x) = x^2\) and \(f(x) = 2x + 1\). 2. Simplify the num

1. We are asked to prove that for all $n \in \mathbb{N}$, the sum $\sum_{k=0}^{n-1} (2k+1)$ equals $n^2$. 2. Rewrite the sum explicitly to understand it better:

1. **Problem 1**: Determine if there exist non-negative quantities $x_1, x_2, x_3$ of Products 1, 2, 3 such that the labor hours in departments A, B, C fully use the monthly capaci

1. Problem 17 (a): Simplify $\sqrt{98}$. We factor 98 as $98 = 49 \times 2$.

1. State the problem: Solve the equation $4x - 28 = 0$ using the quadratic formula. 2. Rearrange the equation as a quadratic form $ax^2 + bx + c = 0$. Since there is no $x^2$ term,

### Exercise 03: Prove using mathematical induction #### 1. Prove that for all $n \in \mathbb{N}$, $\sum_{k=0}^{n-1} (2k+1) = n^2$.