🧮 algebra

1. **State the problem:** We are given a revenue function depending on the number of basic subscribers $c$ and premium subscribers $d$. We need to find the price of each subscripti

1. **Problema:** Factorice completamente $w^2 - 2w - 15$. 2. **Fórmula y reglas:** Para factorizar un trinomio cuadrado de la forma $ax^2 + bx + c$, buscamos dos números que multip

1. The problem asks which function represents the graph of $f(x) = e^x$ shifted 3 units horizontally to the right. 2. The rule for horizontal shifts is: shifting a function $f(x)$

1. The problem asks to identify the graph transformation of the function $g(x) = e^{x+1}$ from the graph of $f(x) = e^x$. 2. The general rule for horizontal shifts in functions is:

1. The problem is to sketch the graph of the function $h(x) = \log_2(x + 2)$ and state its domain, range, and asymptote. 2. The logarithmic function $\log_b(x)$ is defined only for

1. **State the problem:** Solve the equation $$\log_9(x - 5) + \log_9(x + 3) = 1$$ for $x$. 2. **Recall the logarithm property:** The sum of logarithms with the same base can be co

1. **State the problem:** Factorise the expression $$14y^2 + 14$$ completely. 2. **Identify the common factor:** Both terms have a common factor of 14.

1. **State the problem:** Factorise the expression $$11y^2 + 11$$ completely. 2. **Identify the common factor:** Both terms have a common factor of 11.

1. **State the problem:** Solve the linear equation $8x - 2y = 5$ for $y$ in terms of $x$. 2. **Formula and rules:** To isolate $y$, we use basic algebraic operations: addition, su

1. Wir sollen die Bruchgleichung $$\frac{1}{x-3} = \frac{2}{x-1}$$ lösen. 2. Wichtig: Bei Bruchgleichungen dürfen Nenner nicht null sein. Also gilt:

1. Problema 84: Calcolare $$\sqrt[3]{9x^{3}} \cdot \sqrt[3]{16x^{6}} + \sqrt{16x^{3}}$$. Formula usata: $$\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{ab}$$ e $$\sqrt{a^{2}} = a$$ se $

1. Problema 36: Calcolare $$\frac{(\sqrt[3]{3} \cdot \sqrt[3]{3})^2}{(\sqrt{3})^3}$$. 2. Formula e regole: \(\sqrt[n]{a} = a^{\frac{1}{n}}\), \(a^m \cdot a^n = a^{m+n}\), \(\frac{a

1. Problema: Semplificare le espressioni date assumendo tutte le variabili non negative. 2. Formula e regole importanti:

1. **Stating the problem:** Simplify the expression $$80 \sqrt{10} + \sqrt{40}$$. 2. **Formula and rules:** Recall that $$\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$$ and that we c

1. **Problem statement:** Find the difference $A - B$ given the equation $$-x^2y + A + 2xy^2 - B = 3x^2y - 4xy^2$$

1. The problem is to evaluate the expression $10^{10^{17.962951986}}$. 2. This is a power tower or tetration expression where the exponent itself is an exponential number.

1. **State the problem:** We want to solve the sum $\frac{1}{2} + \frac{1}{2} + \frac{1}{4}$.\n\n2. **Formula and rules:** To add fractions, they must have a common denominator. If

1. The problem is to understand the equation $$X^3 + Y^3 + Z^3 = K$$ and analyze its components. 2. This is a cubic equation involving three variables $X$, $Y$, and $Z$, and a cons

1. **Problem statement:** Simplify the expression \(\frac{5 \cdot 4^{15} \cdot 9^9 - 4 \cdot 3^{20} \cdot 8^9}{5 \cdot 2^9 \cdot 9^{16} - 7 \cdot 2^{29} \cdot 27^6}\). 2. **Recall

1. **State the problem:** Simplify the expression $I(x) = \frac{2x^2 + 5x}{x + 1}$. 2. **Formula and rules:** To simplify a rational expression, factor the numerator and denominato

1. **Planteamiento del problema:** Se tiene la función de ingresos anuales en millones de dólares en función de la inversión en I+D (x en millones):