🧮 algebra

1. **State the problem:** We need to compare $A = n$ and $B = 12$ given the equation $n^{11} = 11^n$. 2. **Analyze the equation:** The equation $n^{11} = 11^n$ means the number $n$

1. Diberikan sistem persamaan dengan bilangan real positif $a$ dan $b$: $$a - 12b = 11 - \frac{100}{a}$$

1. **State the problem:** Given the equation $x^{-1} = \frac{\sqrt{2}}{6}$, find the value of $x$. 2. **Recall the rule:** $x^{-1}$ means the reciprocal of $x$, so $x^{-1} = \frac{

1. **Problem statement:** Simplify the algebraic expressions given: A = (4x - 1)(5x - 3)(4x - 1)

1. Planteamos el problema: Encontrar los coeficientes $a$, $b$ y $c$ de la parábola $y = ax^2 + bx + c$ que pasa por el punto $(1,2)$ y es tangente a la recta $y = x$ en el origen

1. **State the problem:** Simplify the expression $$\frac{\frac{1}{8} + \frac{3}{40}}{\frac{1}{8}}$$ and compare it to the fraction $$\frac{9}{5}$$. 2. **Find a common denominator

1. **State the problem:** Simplify the given complex algebraic expression involving variables $t_1$, $t_2$, $r_1$, $r_2$, $\epsilon_1$, $\epsilon_2$, and $x$. 2. **Identify the exp

1. The problem is to find the quadratic equation for the expression $x^2 - 2x - 8 = 0$. 2. The quadratic equation is generally written as $ax^2 + bx + c = 0$, where $a$, $b$, and $

1. **State the problem:** Find the coefficient of $n$ in the expression $$(4n+3)(n+5) - (2n-3)^2.$$\n\n2. **Expand each part:**\n\nFirst, expand $$(4n+3)(n+5)$$ using distributive

1. **State the problem:** We are given the length of a path on a map as 2 cm and the real length of the path as 700 m. We need to find the ratio of the length on the map to the rea

1. **State the problem:** We need to find which petrol is better value for money between Kieran's and Sophie's petrol, and then calculate the cost of 20 litres from the cheaper pet

1. The problem is to compare the cost per litre of petrol for Kieran and Sophie to determine who pays more per litre. 2. The formula to find the cost per litre is:

1. Let's start with **algebra**. Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It helps us solve equations and understand re

1. **State the problem:** We need to express the quadratic expression $4x^2 - 12x + 13$ in the form $(2x + a)^2 + b$, where $a$ and $b$ are constants. 2. **Recall the formula:** Th

1. **Problem:** Prove by mathematical induction that for every positive integer $n$, $$5 + 7 + 9 + \ldots + (2n + 3) = n(n + 4)$$

1. **State the problem:** We are given parameters $t=3$, $k=18$, $\eta=-3$, weights $w=\{\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\}$, $n=3$, matrices $x$ and $P$ with given values, an

1. **State the problem:** Factorize the expression $ax + ay + bx + by$. 2. **Group terms:** Group the terms to factor by common factors:

1. **Problem:** When a number is multiplied by 8 and 12 is added to it, the result is 60. Find the number. 2. **Formula and setup:** Let the number be $x$. The problem translates t

1. **State the problem:** Simplify the expression $$\frac{1 + \frac{1}{x}}{x - \frac{1}{x}}$$. 2. **Rewrite the expression:** To simplify, write the numerator and denominator with

1. Problem: Find the 50th term of the sequence defined by $4n+3$ given the 100th term is 403. 2. Formula: The $n^{th}$ term of the sequence is given by $T_n = 4n + 3$.

1. **Problem:** If the $100^{th}$ term of the sequence defined by $4n+3$ is 403, find the $50^{th}$ term. 2. **Formula:** The $n^{th}$ term of the sequence is given by: