🧮 algebra

1. Problem: Given $z_1 = 2(\cos 45^\circ + i\sin 45^\circ)$ and $z_2 = 3(\cos 120^\circ + i\sin 120^\circ)$, find $z_1.z_2$ in polar form and in $a+bi$ form. Step 1: Multiply modul

1. Stating the problem: Simplify the expression $$\frac{(4^n+1+4^n)^2}{(2^{2n+1}-2^{2n})^2}$$. 2. Simplify the numerator: Combine like terms inside the parentheses.

1. **State the problem:** Find the limit $$\lim_{x\to \infty}\frac{5^{x+1} + 7^{x+1}}{5^x - 7^x}$$ without using differentiation or L'Hopital's Rule.

1. Stating the problem: Solve the equation $10 = \sqrt{x} 2x^{2}$ for $x$. 2. Rewrite the equation clearly: $10 = 2x^{2} \sqrt{x}$.

1. **Problem statement:** We have a line with equation $$y = -5x + k + 5$$ and a curve with equation $$y = 7 - kx - x^2$$.

1. The problem is to find the solutions of an equation using the appropriate formula. 2. For example, to solve a quadratic equation $ax^2 + bx + c = 0$, we use the quadratic formul

1. The problem is to multiply two matrices and simplify the result. 2. The first matrix is $$\begin{bmatrix}-2 & 0 & 5\\ -1 & 3 & 6\\ 0 & 1 & -1\end{bmatrix}$$ and the second matri

1. The problem is to sketch the graph of the function $$y = |3x| + 2$$ using symmetry and intercepts. 2. Identify key features:

1. **State the problem:** Simplify the expression $$\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{4}+\sqrt{3}}$$. 2. **Rationalize each denominator separately:**

1. **Problem B:** Simplify $$8 - \{-9 + 6 - [ +3 + 8 - (-7 + 3) - 9 + 2] - 7 + 3\}$$ Step 1: Simplify innermost brackets $$(-7 + 3) = -4$$

1. نُعطي الدالة: $$f(x)=\frac{x+3}{x-1}$$. 2. لإيجاد خطوط التقارب العمودية، ننظر إلى قيم x التي تجعل المقام يساوي صفرًا لأن الدالة غير معرفة عندها.

1. Problem 1a: Write the set notations for shaded regions. ai) The shaded region covers sets R and Q but not the third set, so the union of R and Q minus the third set: $$R \cup Q\

1. لفهم ما تعنيه ب"لا اقصد الحل تاع الداله"، يُرجى توضيح سؤالك أو ما تحتاجه بالضبط حول الدالة. 2. هل ترغب في معرفة خصائص الدالة مثل المجال، المدى، تقاطع المحاور، نقاط القمة أو القا

1. Stating the problem: Simplify the expression $$1\sqrt{x}2+1+1/\sqrt{x}3+\sqrt{x}2+1/\sqrt{x}4+\sqrt{x}3$$ and write it in a clearer algebraic form. 2. First, clarify the express

1. We are given the function $f(x) = \frac{\sqrt{x-1}}{x+3}$. We need to analyze it. 2. First, consider the domain. The expression inside the square root, $x-1$, must be non-negati

1. The problem states that \(g(x) = f(x) + 2\) and asks to find the domain of \(g\). 2. To find the domain of \(g\), we need to understand the domain of \(f\) because adding 2 to \

1. **State the problem:** Calculate the weighted average price per unit of items purchased and explain why the weighted average is more meaningful than the simple average in this c

1. سأنعطيك مجموعة من التمارين المقترحة في مادة الجبر (Algebre 1) لمساعدتك على التحضير لاختبار CC1. 2. التمرين الأول: حل المعادلة $2x + 5 = 17$.

1. **State the problem:** Arrange the given values from least to greatest. The values are:\n\n√9, 10/13, \int_2^5 x \, dx, 6!, \frac{4\pi}{6}, \log_3(7), \infty, e^2, \sum_{i=3}^8

1. **State the problem:** Evaluate the expression $-1.5 + \left( \frac{9}{10} \div 0.25 \right) - \left( 0.75 + \frac{5}{6} \right)$. 2. **Calculate the division inside the first p

1. **Problem:** Find the set of values of $x$ for which $(x + 7)(x - 4) \leq 0$. Step 1: Identify critical points where the expression equals zero: $x + 7 = 0 \Rightarrow x = -7$,