🧮 algebra

1. **Problem Statement:** A total sum of 700 is to be distributed as seven cash prizes such that each prize is 20 less than the previous one. We need to find the value of each priz

1. **State the problem:** Simplify or express the sum $a^4 + b^4$ in a useful form. 2. **Formula and rules:** There is no simple factorization over real numbers for $a^4 + b^4$ lik

1. The problem is to find the value of $x$ in the equation $$\frac{2x+3}{4} = 5.$$\n\n2. The formula used here is to solve linear equations by isolating the variable $x$.\n\n3. Fir

1. **Problem statement:** Simplify the expression $$\frac{5^{m+3} + 125 \cdot 5^{m-1}}{5^{m+4} \div 25}$$. 2. **Recall important rules:**

1. **State the problem:** Solve the equation $$\sqrt{3} + 2x = -x$$ for $x$. 2. **Rewrite the equation:** Move all terms involving $x$ to one side:

1. The problem is to analyze the summation $$\sum_{i=1}^n iL(2)^{x+2} = (n-2)^{x+2} A$$ with the condition $$n > 0$$. 2. First, understand the summation notation: $$\sum_{i=1}^n iL

1. **Stating the problem:** We want to verify or understand the formula for the sum $$\sum_{i=1}^{n+1} i(2^i) = n \cdot 2^{n+2} + 2$$. 2. **Recall the formula for the sum of the fo

1. **State the problem:** Solve the quadratic equation $2x^2 - 13x + 20 = 0$. 2. **Formula used:** The quadratic formula is given by

1. **Problem:** Find the value of $32 \equiv \, ? \pmod{12}$. 2. **Formula:** To find $a \equiv r \pmod{m}$, compute the remainder $r$ when $a$ is divided by $m$.

1. The problem is to verify the factorization of the quadratic expression $$2x^2 - 13x + 20$$ given as $$(2x - 5)(x - 4)$$. 2. The formula for expanding two binomials is:

1. The problem is to simplify the expression where the first \(a\) is on the 2 and the second \(a\) is on the top with 3, which suggests the expression is \(a^{2a^3}\). 2. The expr

1. **State the problem:** Simplify the expression $$-\frac{1}{2}a + \frac{3}{2}a - 4$$. 2. **Identify like terms:** The terms $$-\frac{1}{2}a$$ and $$\frac{3}{2}a$$ are like terms

1. **State the problem:** Simplify the expression $\frac{5x}{5+x} + 17.5$. 2. **Understand the expression:** The expression consists of a rational term $\frac{5x}{5+x}$ and a const

1. The problem is to simplify the expression $\frac{5x}{5} + x + 17.5$. 2. Use the rule that $\frac{a \cdot b}{a} = b$ when $a \neq 0$.

1. **State the problem:** Find the value of $-32 \equiv ? \pmod{12}$. 2. **Recall the definition:** For integers $a$ and $m > 0$, $a \equiv r \pmod{m}$ means $r$ is the remainder w

1. **State the problem:** Find the equation of the line passing through points $(-8,7)$ and $(0,-1)$ in slope-intercept form $y=mx+b$. 2. **Formula:** The slope $m$ of a line throu

1. **Stating the problem:** Calculate the value of the expression $$\sqrt{16} \times 3 \times \sqrt{7} \times 1 \times \sqrt{25} \times 12 \times 7 \times \sqrt{36} \times 2 \times

1. Problema a resoldre: Calcula i simplifica l'expressió $$3 - 5 \times \frac{1}{4} - 2 \div 8$$. 2. Fórmules i regles importants:

1. The problem is to simplify the expression $+17 + (+1,7)$. 2. First, note that the expression contains a comma in $(+1,7)$ which is not standard notation for a number. Assuming i

1. The problem is to find the sum of $-8$ and $-27$. 2. We use the rule for adding integers with the same sign: add their absolute values and keep the common sign.

1. The problem is to find the sum of $-7$ and $-24$. 2. When adding two negative numbers, you add their absolute values and keep the negative sign.