🧮 algebra

1. Problema 40: Considera la matriz $$A = \begin{pmatrix} 1 & \frac{1}{8} & \frac{1}{8} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

1. Planteamos el problema: Calcular $A^{10}$ para la matriz $$A=\begin{pmatrix}0 & a & -b \\ 0 & 0 & b \\ 0 & 0 & 0\end{pmatrix}$$ donde $a,b$ son constantes. 2. Observamos que $A$

1. The problem asks to find the value of the expression $-6k + 5$ when $k = -3$. 2. The formula given is $-6k + 5$. We substitute $k$ with $-3$.

1. The problem asks to find which card matches Card C's simplified expression $x^{-15}$. 2. Card C is $\frac{(x^6)^{-2}}{x^3}$.

1. **State the problem:** Find the greatest common factor (GCF) of the numbers 48 and 136. 2. **Prime factorization:**

1. **State the problem:** Solve the equation $93 + 2k + 96 = 6(-7k + 3) + 35$ for $k$. 2. **Combine like terms on the left side:**

1. **State the problem:** Find the greatest common factor (GCF) of the numbers 12 and 28. 2. **Recall the formula and rules:** The GCF of two numbers is the product of the lowest p

1. **State the problem:** Solve for $k$ in an equation involving $k$ (the exact equation is not provided, so we assume a general approach). 2. **General approach:** To solve for $k

1. **State the problem:** Expand the logarithmic expression $$\ln \left( \frac{x^3}{\sqrt{x^2 - 36}} \right)$$ using the laws of logarithms, given that $$x > 6$$. 2. **Recall the l

1. **State the problem:** Expand the logarithmic expressions using the laws of logarithms. 2. **Recall the laws of logarithms:**

1. **Problem Statement:** We need to sketch the graph of the function $y=3x+2$ and identify if it is linear.

1. **Problem Statement:** Sketch the graph of the function $y=3x+2$ and identify if it is linear.

1. **State the problem:** Solve the equation $$-10x + 2x = -72$$ for $x$. 2. **Combine like terms:** On the left side, combine $-10x$ and $2x$:

1. Evaluate: a) $\frac{1}{4} + \left(-\frac{3}{4}\right)$ Start by adding the fractions:

1. **Énoncé du problème :** Trouver le point d'intersection des droites du système : $$3x - 2y = 12$$

1. **State the problem:** Solve the equation $$-r - \frac{10}{2} = 9$$ for $r$. 2. **Simplify the fraction:** $$\frac{10}{2} = 5$$, so the equation becomes $$-r - 5 = 9$$.

1. **Énoncé du problème :** Trouver le point d'intersection des droites définies par les équations $$3x - 2y = 12$$ et $$x - 2y = 8$$. 2. **Formule et méthode :** Pour résoudre gra

1. The problem is to find the coordinates where two curves intersect. 2. To find the intersection points of two functions $y=f(x)$ and $y=g(x)$, we set them equal: $$f(x) = g(x)$$

1. **State the problem:** Simplify the expression $X^2 - a^2 - y^2 - 2ay$. 2. **Recall the formula:** Recognize that $y^2 + 2ay + a^2$ is a perfect square trinomial equal to $(y +

1. **State the problem:** Factor the expression $$2x^5 + 14x^3 + 20x$$. 2. **Identify the greatest common factor (GCF):** Look at the coefficients 2, 14, and 20. The GCF of these n

1. **State the problem:** We want to find the predicted value of a car in 2016, given it depreciates exponentially from $23,000 in 2008 to $6,400 in 2013. 2. **Formula for exponent