🧮 algebra

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

1. Problem: Simplify the expression $$\sqrt{6}-\sqrt{\frac{7}{2}+\frac{\sqrt{12}+\sqrt{27}}{\sqrt{300}}}$$. 2. Simplify inside the square roots:

1. El problema es expandir el binomio $$ (x+y)^5 $$ usando el teorema del Binomio de Newton. 2. El teorema del binomio nos dice que:

1. لنفترض أنه نريد تبسيط التعبير الجبري: $f(2) = 2 + x$ أو $2 - x$. 2. بدايةً، إذا كان المقصود هو دالة تعتمد على $x$ مثل $f(x) = 2 + x$ أو $f(x) = 2 - x$، فهذه دوال خطية بسيطة.

1. The problem is to simplify the expression $$\sqrt{6} - \sqrt{\frac{7}{2}} + \frac{\sqrt{12} + \sqrt{27}\sqrt{300}}{}$$. 2. Simplify each square root:

1. Stated problem: Solve for $n$ in the equation $$\frac{11}{7} = \left(\frac{101}{100}\right)^n.$$\n2. Start by taking the natural logarithm of both sides to deal with the exponen

1. We are given an estimated area of 785 m² and the block dimensions of 30 m and 26 m. We need to calculate the actual area and find the percentage error in the estimation. 2. Calc

1. State the problem: Solve for $n$ in the equation $$10^n = 7 \times 11^{n-1}$$. 2. Rewrite the right side to isolate the exponential term: $$10^n = 7 \times \frac{11^n}{11} = \fr

1. **State the problem:** Given an arithmetic sequence with the last term $a_n = 4$, common difference $d = 2$, and sum of $n$ terms $S_n = -14$, find $n$ (the number of terms) and

1. Stating the problem: Sydney and Ann invested a total of 232212 in the ratio 1:2, find how much Ann invested. 2. Let Sydney's investment be $x$.

1. The problem is to convert decimals to percents. 2. To convert a decimal to a percent, multiply it by 100 and add the percent symbol (%).

1. The problem is to simplify the expression $\frac{X}{x-6} - \frac{5}{2}$. 2. To combine the two terms, find a common denominator, which is $2(x-6)$.

1. Stated the problem: Given the function $m(x) = x^2 - 4x + 3$ with domain $x \geq p$, find the value of $p$ so that $m$ has an inverse function. Also, given $g^{-1}h(x) = 5x + 4$