🧮 algebra

1. The problem is to compute the value of $22^5$. 2. Recall that $22^5$ means $22$ multiplied by itself $5$ times: $$22^5 = 22 \times 22 \times 22 \times 22 \times 22$$.

1) **Determine whether the sequence converges or diverges, and if it converges, find its limit.** **a)** $a_n = \frac{3 + 5n^2}{n + n^2}$

1. State the problem: Simplify the expression $$ (2a + 2b)^2 + (2a - 2b)^2 $$. 2. Expand each square term using the formula $$(x + y)^2 = x^2 + 2xy + y^2$$ and $$(x - y)^2 = x^2 -

1. State the problem: Simplify $|x|^3$. 2. Interpretation: I read this as the cube of the absolute value, which is $|x|^3$.

1. Stating the problem: Calculate the total cost for each raw material given the cost per kg and amount used per year, then find the total cost of all materials combined. 2. Calcul

1. Let's start by defining the problem: Polynomial division is a method to divide one polynomial (the dividend) by another polynomial (the divisor), similar to long division with n

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

1. **Statement of Problem**: Given two propositions $P$ and $Q$: - $(P): \exists x \in \mathbb{R} : x^2 + x - 2 = 0$

1. The original problem had \( \sqrt{25} \), which equals 5. 2. Now we replace \( \sqrt{25} \) with \( \sqrt{15} \).

1. **Problem 2:** Given functions $f(x) = a + x^2$ and $L(x) = c$ where $a$, $c$ are constants, and the equation $3f(2) + 3L(x) = 6$ holds for all $x$. We need to find $2f(0) + 2L(

1. The problem is to multiply two binomials using the extended method (also known as the FOIL method). Let's consider the binomials $(a+b)$ and $(c+d)$. 2. The extended method mult

1. **Problem 9 (a):** Find the natural domain and range of $f(x)=\frac{1}{x-3}$. Domain: The denominator cannot be zero, so $x-3\neq0 \Rightarrow x\neq3$.

1. We are asked to simplify the expression $$\sqrt{\frac{2-\sqrt{3}}{2+\sqrt{3}}}+\sqrt{\frac{2+\sqrt{3}}{2-\sqrt{3}}}$$. 2. Start by rationalizing the denominators inside each squ

1. Problem: Find the natural domain and range of (a) $g(x)=\frac{x+1}{x-1}$ and (b) $g(x)=\begin{cases} \sqrt{x+1}, & x\geq 1 \\ 3, & x<1 \end{cases}$. 2. For (a) $g(x)=\frac{x+1}{

1. We start by identifying the problem: explain every step in exercise 9 and 10. 2. Since the exercises are not fully specified, I'll assume exercise 9 and 10 involve typical algeb

1. The problem is to solve the equation $2x + 3 - 1 = 5\frac{4}$. 2. First, simplify the left side: $2x + (3 - 1) = 2x + 2$.

1. Stating the problem: Solve the algebraic equation $2x + 3 - 1 = 5$. 2. Simplify the left side by combining like terms: $3 - 1 = 2$, so the equation becomes $2x + 2 = 5$.

1. **State the problem:** Solve the system of equations: $$x+2y=6$$

1. **Stating the problem:** Simplify the expression $$1+\frac{1}{2+\frac{1\sqrt{2}}{1+2}}$$. 2. **Simplify the denominator inside the fraction:** The denominator inside the inner f

1. Problem 3: Determine if the graph defines $y$ as a function of $x$ for each graph. (a) The graph is an increasing curve passing through the origin resembling a cubic function. I

1. **State the problem:** Machine A can finish the job in 4 hours, and Machine B can finish the same job in 6 hours. We want to find how long it takes for both machines working tog