🧮 algebra

1. **State the problem:** We need to evaluate the expression $$Tr = \frac{5^2}{\pi^2 \times \alpha} \ln \left[ \frac{8}{\pi^2} \times \frac{240 - 60}{105 - 60} \right]$$ where $\al

1. **State the problem:** We need to simplify and evaluate the expression $$T_r = \frac{5^2}{\pi^2 \times \alpha} \ln \left[ \frac{8}{\pi^2} \times \frac{240-60}{105-60} \right]$$

1. **Problem statement:** Mary's grandparents invest an amount $x$ on 1st January 1998 and continue to deposit $x$ on the first day of each month. The account pays 0.4% interest pe

1. **State the problem:** Solve the equation $$\log_3 \sqrt{x} = \frac{1}{2 \log_2 3} + \log_3 (4x^3)$$ where $$x > 0$$. 2. **Recall logarithm properties:**

1. **Problem Statement:** We need to clarify the expression and solve or simplify it based on the correction: the term is $4x^3$ at the end, not $2$, and there is no $x$ after the

1. **State the problem:** Solve the equation $$\log_3 \sqrt{x} = \frac{1}{2 \log_3 x} + \log_3 (4x^2)$$ where $$x > 0$$. 2. **Recall logarithm properties:**

1. **State the problem:** Given the value of $n$ as 14, we want to understand or use this value in a mathematical context. 2. **Explanation:** Since the problem only states "n is 1

1. **State the problem:** We have two numbers, 27 and 36, and we want to show that their least common multiple (LCM) is 108 by grouping blocks in tables of 108 blocks. 2. **Recall

1. **State the problem:** We are given the midpoint $M(-3,-1)$ of segment $\overline{AB}$ and one endpoint $A(-2,5)$. We need to find the coordinates of the other endpoint $B(x,y)$

1. **State the problem:** We are given the midpoint $M(-4,4)$ of the line segment $\overline{AB}$ and one endpoint $A(-6,1)$. We need to find the coordinates of the other endpoint

1. **State the problem:** Simplify the expression $$\frac{5ef \cdot y}{10ef - 5fg}$$. 2. **Identify the formula and rules:** We will factor the denominator and simplify the fractio

1. مسئله را بیان می‌کنیم: تابع $f(x) = x + \sqrt{x^3 + 1}$ داده شده است و وارون آن به صورت $f^{-1}(x) = ax + \frac{b}{x}$ برای $x > 0$ است. هدف یافتن حاصل $ab$ است. 2. برای یافتن و

1. **State the problem:** We are given the midpoint $M(-2,-3)$ of the line segment $\overline{AB}$ and one endpoint $A(-8,-5)$. We need to find the coordinates of the other endpoin

1. مسئله را بیان می‌کنیم: باید مقدار $$g^{-1}(f^{-1}(8))$$ را پیدا کنیم که در آن $$f(x) = \frac{4x + 3}{2x - 4}$$ و $$g(x) = \frac{x^2 + 1}{4}$$. 2. ابتدا معکوس تابع $$f$$ را پیدا

1. Let's clarify the problem: You want to understand why, in some calculations, we multiply by 11 and 10 instead of directly dividing $v_{11}$ by $v_{10}$ and then taking the 11th

1. **State the problem:** We have the addition of three 3-digit numbers where $x$ and $y$ are single digits, and the sum is a 4-digit number: $$

1. **Problem 1: Identify which is NOT a factor of the polynomial $P(x) = x^3 - \pi x^2 - x + \pi$.** We test each candidate factor by evaluating $P(x)$ at the root of the factor. I

1. The problem is to work with a number represented by $x$ instead of $W$. 2. This means wherever the variable $W$ was used, we now replace it with $x$.

1. **Problem Statement:** A field of paddy can be harvested by 10 people in 20 days. If 2 people are less (i.e., 8 people), how many days will it take to harvest the same field? 2.

1. **Problem 1:** For what number $k$ is $x+1$ a factor of $f(x) = 2x^6 - kx^3 + 5x - 1$? By the Factor Theorem, if $x+1$ is a factor, then $f(-1) = 0$.

1. The first problem asks for the sum of the first four terms of a geometric sequence where the first term $G_1 = \frac{1}{4}$ and the common ratio $r = 3$. 2. The formula for the