🧮 algebra

1. The user presents several calculations involving powers and numbers: 2. The first expression is $5 \times 5 \times 5$, which equals $5^3$. Calculating, $5 \times 5 = 25$ and $25

1. The problem is to evaluate the expression $1 + (-1)^1$. 2. First, calculate the exponentiation: $(-1)^1 = -1$ since any number raised to the power of 1 is the number itself.

1. **State the problem:** We have the matrix equation $$\begin{bmatrix} x \\ y \end{bmatrix} = \frac{1}{5} \begin{bmatrix} 1 & -2 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} 3 \\ 4 \end

1. **State the problem:** Given the equation

1. It seems you want to see the visual of a "through." 2. Assuming you meant the graph of a function looking like a "trough," this corresponds to a quadratic function or a parabola

1. **Problem statement:** Find the value of the determinant $D$ of the matrix $$A=\begin{bmatrix}1 & -2 \\ 2 & 1\end{bmatrix}.$$ 2. The determinant of a $2\times 2$ matrix $$\begin

1. **State the problem:** We are given the system of equations: $$-7x + 2y = 12$$

1. Let's recall that a matrix is lower triangular if all entries above the main diagonal are zero. 2. For matrix (a):

1. The problem asks us to identify the position of the value -17 in the matrix resulting from adding two given 3x3 matrices. 2. We start by adding the matrices element-wise:

1. The problem is to find the determinant of matrix \( D = \begin{bmatrix} 3 & 4 \\ 6 & ? \end{bmatrix} \).\n\n2. Since the user provides elements 3, 4, and 6, we assume the matrix

1. **State the problem:** We need to multiply a 1x3 matrix $A = [65\ 80\ 30]$ by a 3x2 matrix $B = \begin{bmatrix}1 & 10 \\ 2 & 15 \\ 8 & 55\end{bmatrix}$.\n\n2. **Recall matrix mu

1. Stating the problem: Simplify the expression $$\frac{3}{4x} - \frac{5}{6x^2}$$. 2. Find the common denominator: The denominators are $$4x$$ and $$6x^2$$. The least common denomi

1. Simplify $\frac{3}{4}x - \frac{5}{6}x^2$. - The terms share no common factors, so the expression remains as is: $$\frac{3}{4}x - \frac{5}{6}x^2$$.

1. **Problem:** Determine the speeds of accessing files in the Finance bucket folders by year and by month after 4 days. 2. **Given functions:**

1. Let's first clarify what the number 5 represents in the given context. 2. If 5 is part of an equation or expression, please provide the full problem for precise assistance.

1. The problem states we need to find the cost of fencing a garden with given dimensions and a cost per meter. 2. First, identify the perimeter of the garden. The garden dimensions

1. The problem is to understand and express the cube of the variable $g$, which is written as $g^3$. 2. Cubing a number or variable means multiplying it by itself three times: $$g^

1. **State the problem:** We are given the function $$g(x) = 1 + x^2$$ and need to analyze its properties. 2. **Identify the type of function:** This is a quadratic function, where

1. The expression provided is $((x^2 - y^2) + 1 (x y^2))$. Let's clarify and rewrite it for proper interpretation: $$x^2 - y^2 + xy^2$$ 2. We can analyze the function $f(x,y) = x^2

1. **State the problem:** We have two investments that pay out every 4 years and every 6 years respectively. We want to find out after how many years both investments will pay out

1. **State the problem:** Simplify the expression $$(2y - 11)(y^2 - 3y + 2).$$ 2. **Apply distributive property (FOIL):** Multiply each term in the first polynomial by each term in