🧮 algebra

1. **State the problem:** Evaluate the expression $$\sqrt{(-3)^2} + \sqrt{| -4 |}$$. 2. **Recall the rules:**

1. **State the problem:** Solve the equation $-2n + 32 = -4(8n - 8)$ for $n$. 2. **Apply the distributive property:**

1. **State the problem:** Solve the equation $1 + 7(-7k + 5) = -111$ for $k$. 2. **Apply the distributive property:** Multiply 7 by each term inside the parentheses:

1. **Problem statement:** Find the values of $k$ for which each quadratic equation has two distinct roots. 2. **Key concept:** A quadratic equation $ax^2 + bx + c = 0$ has two dist

1. **State the problem:** Solve the quadratic equation $$4x^2 - 5 = 4(x - 5)$$. 2. **Rewrite the equation:** Expand the right side:

1. El problema nos pide expresar varios logaritmos en términos de $a=\log 2$, $b=\log 3$ y $c=\log 5$. 2. Recordemos las propiedades importantes de los logaritmos:

1. O problema pede para fatorar o polinômio $$4x^4 - 13x^3 - x^2 + 13x - 2$$ usando a regra de Ruffini. 2. A regra de Ruffini é usada para dividir polinômios por binômios da forma

1. **State the problem:** Simplify the expression $4x^4 - 13x^3 - x^2 + 13 - 2$. 2. **Combine like terms:** The constants $13$ and $-2$ can be combined.

1. The problem is to evaluate $\log 10$. 2. Recall that $\log$ without a base usually means $\log_{10}$, the logarithm base 10.

1. The problem asks to create a piecewise function that includes at least one absolute value and then graph it. 2. A piecewise function is defined by different expressions dependin

1. **State the problem:** We are given the function $$F(x) = a + \frac{b}{x+1} + \frac{c}{x-3}$$ and we want to understand its structure and behavior. 2. **Formula and rules:** Thi

1. **State the problem:** Simplify the expression $$\frac{3}{a^2-16} + \frac{2}{a+4}$$. 2. **Recall the formula and rules:** The denominator $a^2-16$ is a difference of squares, wh

1. **State the problem:** Simplify the expression $$\frac{4}{y^2-4} + \frac{1}{y-2}$$. 2. **Identify the formula and rules:** Recognize that $y^2-4$ is a difference of squares, whi

1. Planteamos el problema: dividir el polinomio $$4x^4 - 3 + 6x^2 - 852$$ entre el binomio $$x - 4$$. 2. Ordenamos el polinomio dividendos por grados descendentes y completamos tér

1. Problema: Reducir a un solo logaritmo las expresiones dadas. 2. Regla general: Para combinar logaritmos con la misma base, usamos las propiedades:

1. **State the problem:** Simplify the expression $$\frac{1}{2 - \frac{1}{3}}$$. 2. **Recall the rule:** To simplify a complex fraction, first simplify the denominator.

1. **Énoncé du problème** : Trouver l'expression de la fonction $f$ définie par $f(x) = \frac{1}{x} + 1$ et calculer $f(a)$ et $f(b)$ pour $a=1$ et $b=3$. 2. **Formule utilisée** :

1. Problem: Oblicz objętość prostopadłościanu o krawędziach długości $2 - \frac{7}{5}$, $\sqrt{2}$ oraz $\frac{1}{2 - \frac{1}{5}}$. 2. Wzór na objętość prostopadłościanu to:

1. **Problem statement:** Factor the expression $zac - bc$ using grouping. 2. **Formula and rules:** Grouping involves factoring common terms from parts of the expression.

1. **State the problem:** Calculate the new hourly rate given the number of hours worked per week and the new weekly earnings.

1. **State the problem:** Calculate (a)(i) the total hours Mrs. Tracey works per week, (a)(ii) her weekly earnings at $8.10 per hour, and (b) her new hourly rate given new weekly e