🧮 algebra

1. সমস্যাটি হলো: একটি ছিদ্র একটি চৌবাচ্চাকে ৪ ঘণ্টায় খালি করতে পারে। একটি নল প্রতি মিনিটে ৬ লিটার প্রবেশ করায় এবং এখন চৌবাচ্চাটি ১২ ঘণ্টায় খালি হয়। চৌবাচ্চার ধারণ ক্ষমতা কত? 2. প্র

1. **Problem statement:** Find the smallest positive integer $n$ such that $n + 2035$ is a perfect cube. 2. **Understanding the problem:** We want to find $n > 0$ so that $n + 2035

1. **State the problem:** We need to find how many odd numbers between 1 and 2025 are not divisible by 5. 2. **Identify the total odd numbers from 1 to 2025:** Odd numbers are ever

1. Задача: Найти значение выражения $$\sqrt{5+\sqrt{24}} + \sqrt{5-\sqrt{24}}$$. 2. Формула и идея: Чтобы упростить выражение вида $$\sqrt{a+\sqrt{b}} + \sqrt{a-\sqrt{b}}$$, можно

1. The problem asks for the 104th digit after the decimal point of the fraction $S = \frac{2222}{7777}$. 2. To find digits after the decimal point of a fraction, we perform long di

1. **State the problem:** The temperature in city A is two degrees below zero, which means $-2$ degrees. 2. **Identify the temperature in city B:** It is five times warmer than cit

1. **State the problem:** Simplify the expression $$(6\sqrt{10} - 8\sqrt{2})(\sqrt{12} + 2\sqrt{10})$$. 2. **Recall the distributive property:** To multiply two binomials, use $$(a

1. The problem is to check if substituting $x=3$ into an expression equals 128. 2. Since the expression is not given, let's assume the expression is $f(x)$ and we want to verify if

1. **State the problem:** Solve the equation $$4^x + 4^x = 128$$ for $x$. 2. **Combine like terms:** Since both terms are $4^x$, we can write:

1. **State the problem:** Simplify the expression $5\sqrt{45} - 11\sqrt{20} + \sqrt{12} - \sqrt{75}$.\n\n2. **Recall the rule:** $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ and

1. The problem is to verify if the equation $$\sqrt{3} + 2\sqrt{2}\sqrt{3} + \sqrt{2} = \sqrt{5} + \sqrt{24}$$ is correct. 2. First, simplify each term on the left side:

1. **State the problem:** We want to verify if the equation $$\sqrt{3} + 2\sqrt{2}\sqrt{3} + \sqrt{2} = \sqrt{5} + \sqrt{24}$$ is correct. 2. **Simplify the left side:**

1. **مسئله اول: اجتماع، اشتراک و تفاضل بازه‌ها** (الف) اجتماع $(-\infty, 3] \cup [0, 8]$ یعنی همه اعداد که در یکی از این دو بازه هستند.

1. Let's state the problem: We want to check if $\sqrt{3} + 2\sqrt{2}\sqrt{3} + \sqrt{2} = \sqrt{24}$ is correct. 2. Recall the properties of square roots: $\sqrt{a}\sqrt{b} = \sqr

1. **State the problem:** Solve the equation $3(14n+3) - 2(21n + 4) = 1$ for $n$. 2. **Apply the distributive property:** Multiply inside the parentheses:

1. **Problem statement:** Given that $y$ varies directly with $(x-1)^2$, and when $x=4$, $y=18$, find: i. The relationship between $x$ and $y$

1. Problem: Check for mistakes in the given algebraic expansions and factorizations. 2. We will verify each item by expanding or factoring and comparing with the provided expressio

1. **State the problem:** We have a piecewise function $$g(x) = \begin{cases} x - 2 & x \leq 3 \\ x + 1 & 3 \leq x \leq 5 \end{cases}$$

1. The problem is to understand why we multiply by the conjugate of the numerator instead of the denominator when simplifying expressions involving radicals. 2. The conjugate of a

1. **State the problem:** Solve the simultaneous equations: $$2x + 3y = 6$$

1. **Problem statement:** Determine if the function $f(x) = \sqrt{x} + x$ is injective. 2. **Definition:** A function is injective (one-to-one) if for every $x_1, x_2$ in the domai