🧮 algebra

1. Planteamos el problema: calcular el resultado de la operación $$\frac{2a^3b^5}{3} \cdot \frac{-9a^6b}{4}$$. 2. Recordemos que para multiplicar fracciones se multiplican numerado

1. **Problem statement:** Ba số tự nhiên có một chữ số khác nhau được điền vào ba ô vuông dưới cùng: gọi lần lượt là $a$, $b$, $c$ với $a,b,c \in \{0,1,2,...,9\}$ và $a,b,c$ khác n

1. **Problem statement:** Determine the type of decimal representation of the fraction $$\frac{7}{3^{2023} \cdot 2^{2021}}$$. 2. **Recall the rule for decimal representations of fr

1. **Problem statement:** We have a square floor tiled exactly by smaller square tiles of the same size. The total number of tiles on the two diagonals is 45. We need to find the t

1. **Problem:** Test whether $(x - 2)$ is a factor of $f(x) = 2x^3 + 2x^2 - 17x + 10$ and factorize $f(x)$ if it is. 2. **Step 1: Use the Factor Theorem.**

1. **Problem statement:** We need to find the digit $P$ in a five-digit number $PQRST$ where each digit is from 1 to 5, used exactly once, and $P < Q$. 2. **Given conditions:**

1. **Stating the problem:** Given functions $f(x) = x + 3$ and $g(f(x)) = 2x + 5$, find the expressions for $f^{-1}(x)$, $g(x)$, $g(f^{-1}(x))$, and the value of $k$ such that $f(g

1. **Problem statement:** We need to find two consecutive natural numbers $n$ and $n+1$ such that the fractions $\frac{n}{7}$ and $\frac{n+1}{7}$ have the fraction $\frac{9}{11}$ l

1. **Problem statement:** We need to find the sum of three natural numbers given that the sum of their squares is 6525. The second number is \(\frac{2}{3}\) of the first and also \

1. المشكلة: حل معادلة باستخدام التعويضات المثلثية. 2. القاعدة الأساسية: التعويضات المثلثية تستخدم لتحويل تعبيرات تحتوي على جذور أو تعبيرات معقدة إلى تعبيرات مثلثية أبسط.

1. You asked for more questions to prepare for the TOLC test. 2. Since you did not specify a particular topic, here is a sample algebra question to practice:

1. Let's start by understanding the problem: we want to practice elementary operations (addition, subtraction, multiplication, division) and ordering (comparing sizes) between inte

1. **Stating the problem:** Solve the equation $$x^5 - x^4 - x^3 - x + 1 = 0$$ for $x$. 2. **Formula and approach:** This is a polynomial equation of degree 5. There is no general

1. **State the problem:** We are given a pattern where a word with a slash (e.g., X/MAS) corresponds to a fraction (24/247) and then to a decimal (0.097). We need to find the decim

1. **State the problem:** Find the Greatest Common Factor (GCF) of 108 and 100. 2. **Formula and rules:** The GCF of two numbers is the largest number that divides both without lea

1. **Problem:** If the LCM of $m$ and 53 is 1279, find the value of $m$. 2. **Formula and rules:** For two numbers $a$ and $b$, the Least Common Multiple (LCM) and Greatest Common

1. **State the problem:** A phone originally priced at 120000 has increased to 124800. We need to find the percentage increase. 2. **Formula for percentage increase:**

1. **Stating the problem:** We have a car priced at 1800 rials initially, and after two years, its price decreases to 1600 rials. We need to find the percentage decrease in price.

1. **State the problem:** We want to show that $$\sum_{r=3}^n \frac{1}{r(r-2)} = \frac{3}{4} - \frac{1}{2n-2} - \frac{1}{2n}$$

1. **State the problem:** Simplify the expression $$\left(\frac{y}{x^2}\right)^3 \left(\frac{x}{y^2}\right)^2$$. 2. **Recall the power of a quotient rule:** For any nonzero $a$ and

1. **Problem Statement:** Given a geometric series where the 4th term is 10 and the 7th term is 80, find: i. The common ratio $r$