🧮 algebra

1. **Problem Statement:** Verify whether the matrix multiplication $AB$ equals $BA$ for the given matrices: $$A = \begin{bmatrix}1 & 3 \\ 4 & 5\end{bmatrix}, \quad B = \begin{bmatr

1. **Problem Statement:** Solve for $x$ in the equation $x + \log_2(2^x - 6) = 4$. 2. **Formula and Rules:** Recall that $\log_b(a)$ is the logarithm base $b$ of $a$, and the domai

1. **Problem:** Given $a = \sqrt{5 + 2}$ and $b = \sqrt{5 - 2}$, find $a^2 - b^2$. 2. **Formula and rules:** Recall the difference of squares formula: $a^2 - b^2 = (a - b)(a + b)$.

1. The problem is to find the value of the expression or solve the given algebraic problem (please provide the specific problem for detailed steps). 2. Generally, for algebraic pro

1. **State the problem:** Find three-digit numbers such that the number itself is 21 times the sum of its digits. 2. **Define variables:** Let the three-digit number be $100a + 10b

1. **Problem Statement:** Prove that the map $\rho : \mathbb{Z}[i] \to \mathbb{Z}/10\mathbb{Z}$ defined by $\rho(a + bi) = [a + 7b]$ is a homomorphism and find its kernel. 2. **Rec

1. **State the problem:** We want to estimate the solution to the equation $$x^3 + 7x - 5 = 0$$ between $$x=0$$ and $$x=1$$ using the iteration formula $$x_{n+1} = \frac{5}{x_n^2 +

1. **State the problem:** Show that the equation $$x^3 + 7x - 5 = 0$$ has a solution between $$x=0$$ and $$x=1$$. 2. **Evaluate the function at $$x=0$$:**

1. **State the problem:** Write the quadratic expression $x^2 + 6x - 7$ in the form $(x + a)^2 + b$ where $a$ and $b$ are integers. 2. **Recall the formula:** To complete the squar

1. **Problem Statement:** Given a circle with equation $$x^2 + y^2 = 25$$ and a line $$3x + 4y + 1 = 0$$, find:

1. **State the problem:** Yasmin, Kian, and Maggie have scores in the ratio 3 : 5 : 12, and the mean of their scores is 180. We need to find Yasmin's score. 2. **Understand the rat

1. **State the problem:** We are given a table of values for $t$ (days) and $D(t)$, and we want to find which function among the given options models the data correctly. 2. **Given

1. **State the problem:** Solve the equation $$\left(\frac{2}{3}\right)^{x+1} = \left(\frac{16}{9}\right)^{x-5}$$ for $x$. 2. **Rewrite the bases:** Note that $$\frac{16}{9} = \lef

1. **State the problem:** Simplify the expression $$\left(\frac{750}{512}\right)^{\frac{1}{3}}$$ which means finding the cube root of the fraction $\frac{750}{512}$. 2. **Recall th

1. **State the problem:** Simplify the expression $$\frac{a}{a-b} \div \frac{a}{a-c} \div \frac{a}{a-d}$$. 2. **Recall the rule:** Dividing by a fraction is the same as multiplying

1. The problem is to simplify the expression $\frac{a}{a-b} \div \frac{a}{a-c} \div \frac{a}{a-d}$. 2. Recall that dividing by a fraction is the same as multiplying by its reciproc

1. The problem asks for the range of the function represented by the graph. 2. The graph is a parabola opening upwards with its vertex at approximately $(0,1)$.

1. **State the problem:** Solve the equation $x = 2y + x^4$ for $y$ in terms of $x$. 2. **Rewrite the equation:** We want to isolate $y$ on one side. Starting with:

1. The problem gives two functions: $$f(x) = x^2$$ and $$g(x) = (x - 9)^2 + 3$$. 2. We are asked to understand or analyze these functions. Both are quadratic functions, which means

1. The problem is to rewrite the quadratic function $$f(x) = x^{2} + 16x - 46$$ by completing the square. 2. The formula for completing the square for a quadratic $$ax^{2} + bx + c

1. **State the problem:** Simplify the expression $$3(n + 3mn) - 5n(2m - 1)$$. 2. **Use the distributive property:** Multiply each term inside the parentheses by the factor outside