🧮 algebra

1. **Problem Statement:** Find the multiplicative inverse $z^{-1}$ of the complex number $z = \sqrt{3} + i \sqrt{7}$ and prove that $zz^{-1} = 1$. 2. **Formula and Rules:** The mul

1. **State the problem:** Solve the equation $$|3x + 2| + |3x - 2| = 5$$ for $x$. 2. **Understand the absolute value:** The absolute value function $|a|$ equals $a$ if $a \geq 0$ a

1. **State the problem:** Find the coefficient of $x^{12}y^{13}$ in the expansion of $(x+y)^{25}$. 2. **Formula used:** The binomial theorem states that

1. **State the problem:** A student has marks in five subjects: English 73, Urdu 82, Maths 80, History 67, and Science 62. The weights for these subjects are 4, 3, 3, 2, and 2 resp

1. **State the problem:** We are given a principal amount $P = 25000$, interest earned $I = 5000$, and time $T = 4$ years. We need to find the rate of interest $R$. 2. **Formula us

1. مسئله: بررسی و مقایسه نامساوی‌ها و تساوی‌های داده شده شامل لگاریتم‌ها و توانی‌ها است. 2. ابتدا تعاریف و قواعد مهم را مرور می‌کنیم:

1. **Problem:** Identify which statement about the polynomial function $f(x) = -2x + \sqrt{3} x^2 - \frac{1}{2} x^4$ is not correct. 2. **Formula and rules:**

1. **State the problem:** Evaluate the expression $$\frac{(438)^3 \times \sqrt{0.056}}{(388)^4}$$ using logarithms. 2. **Recall the logarithm properties:**

1. **State the problem:** Simplify the expression $$\sqrt{9x^{8}y^{-4}} \times \sqrt[3]{8x^{6}y^{-3}}$$. 2. **Recall the formulas:**

1. **State the problem:** Find the equation of the perpendicular bisector of the line segment joining points $A(0,6)$ and $B(2,-2)$.\n\n2. **Find the midpoint of segment AB:** The

1. **Problem:** Solve the equation $$\sqrt[4]{0.04^x} = 125^{x-2}$$. 2. **Formula and rules:**

1. **State the problem:** Find the value of $x$ such that $\log_2 \left(\frac{1}{128}\right) = x$. 2. **Recall the logarithm definition:** $\log_b a = c$ means $b^c = a$.

1. **State the problem:** Simplify the expression $x^2 + 5x + 2x$. 2. **Identify like terms:** The terms $5x$ and $2x$ are like terms because they both contain $x$ to the first pow

1. Задача: Решить уравнение $$\sqrt[4]{0,04^x} = 125^{x-2}$$. 2. Формулы и правила:

1. The problem is to find the value of the square root of 3. 2. The square root of a number $x$ is a value $y$ such that $y^2 = x$.

1. The problem is to find a simple formula for expanding a bracket multiplied by 3. 2. When you have an expression like $3(a + b)$, the distributive property of multiplication over

1. **Problem statement:** Find the remainder when dividing the polynomial $4x^3 + 6x^2 + 3x + 4$ by $2x^2 + 1$. 2. **Formula and rules:** When dividing a polynomial $P(x)$ by a pol

1. **Nêu bài toán:** Tìm số các số tự nhiên $n$ sao cho biểu thức $$\frac{4x^5 y^2 - 3x^3 y + x^2 y}{3x^n y}$$ chia hết, tức là kết quả là một đa thức không có mẫu số. 2. **Phân tí

1. Bài toán yêu cầu tìm giá trị của $x$ sao cho phương trình $$x^3 - x^2 + \frac{1}{3}x - \frac{1}{27} = 0$$ được thỏa mãn. 2. Ta nhận thấy đây là phương trình bậc ba, có thể thử n

1. **State the problem:** Simplify the expression $$\frac{x^2}{x-3} - \frac{9}{x-3}$$. 2. **Identify the common denominator:** Both terms have the same denominator $x-3$, so we can

1. **State the problem:** Solve the linear equation $-3.25x + 6.31 = 10.52 + 5.42x$ for $x$. 2. **Formula and rules:** To solve for $x$, we want to isolate $x$ on one side of the e