🧮 algebra

1. **State the problem:** We need to find the commission earned by an employee based on total revenue, sales, and cancellations. 2. **Given data:**

1. The problem asks to insert two arithmetic means between 8 and 20. 2. An arithmetic mean between two numbers is a number that forms an arithmetic progression (AP) with the given

1. **State the problem:** Find the sum of the first 20 terms of an arithmetic progression (AP) where the first term $a_1=7$ and the common difference $d=3$. 2. **Formula used:** Th

1. Problem: Calculate the value of the expression $3 \times \frac{2}{5} + \frac{2}{3}$. 2. Formula and rules: To add fractions, they must have a common denominator. Multiplication

1. Stating the problem: Simplify the expression $$\left(\frac{2}{5}\right)^{-3} \times \left(\frac{2}{5}\right)^3 + \left(\frac{3}{5}\right)^5 \times \left(\frac{3}{5}\right)^{-5}.

1. **State the problem:** We need to find the amount owed after 10 years on a loan of 2000 at an 18% annual interest rate, compounded quarterly. 2. **Formula used:** The compound i

1. **State the problem:** We are given a line $p$ with slope $m = -\frac{5}{3}$ and an $x$-intercept at $(-6, 0)$. We need to find the $y$-coordinate of the $y$-intercept of this l

1. **State the problem:** We are given a linear function $f(x)$ and its graph, and we want to find which equation form $f(x)$ could have, given that $c$ and $d$ are positive consta

1. The problem is to solve the equation $2x + 3 = 7$ for $x$. 2. We use the basic algebraic principle of isolating the variable $x$ by performing inverse operations.

1. **Problem statement:** Find the real values of $X$ and $Y$ that satisfy the equation $$X(X+i) + Y(Y - i) + i = 13.$$ 2. **Rewrite the equation:** Expand the terms:

1. **State the problem:** Evaluate the expression $16.66 - \left(\frac{1}{4} + 3.15\right)^3$. 2. **Identify the formula and rules:** We need to first simplify inside the parenthes

1. The problem is to solve the equation for $x$: $2x + 3 = 11$. 2. The formula used here is to isolate $x$ by performing inverse operations. Important rules: whatever you do to one

1. **State the problem:** Evaluate the expression $$\frac{1}{\sqrt{2}} + \pi$$ using the approximations $$\sqrt{2} = 1.414$$ and $$\pi = 3.141$$ up to three decimal places. 2. **Re

1. **State the problem:** Simplify and evaluate the expression $$\left( \frac{12^{\frac{1}{5}}}{27^{\frac{1}{5}}} \right)^{\frac{5}{2}}$$. 2. **Recall the exponent rules:**

1. **State the problem:** Calculate the value of the expression $15.66 - \left[\frac{1}{4} + 3.15\right]^2$. 2. **Recall the order of operations:** First, solve inside the brackets

1. The problem is to define what a real root is in mathematics. 2. A real root of an equation is a solution that is a real number. For example, if we have a polynomial equation $f(

1. **State the problem:** We need to express the fraction $$\frac{5 - \sqrt{3}}{2 + \sqrt{3}}$$ in the form $$x + y\sqrt{3}$$ where $x$ and $y$ are rational numbers. 2. **Formula a

1. **Problem statement:** Given the function $k(x) = |x - 3| - |x + 5|$, we need to write it as a simplified piecewise function by considering three intervals. 2. **Intervals to co

1. Diberikan sistem persamaan: $$3x + 2y = \log_4 \left(18^{\frac{5}{4}}\right)$$

1. المشكلة: حل المعادلة $x^2 - 5x + 6 = 0$. 2. الصيغة المستخدمة: معادلة من الدرجة الثانية على الشكل $ax^2 + bx + c = 0$، حيث $a=1$, $b=-5$, و $c=6$.

1. لنبدأ بتعريف الدالة العددية: الدالة العددية هي علاقة تربط كل عنصر من مجموعة تسمى المجال (domain) بعنصر واحد فقط من مجموعة أخرى تسمى المدى (range). 2. الصيغة العامة للدالة العددي