🧮 algebra

1. **Problem Statement:** Solve the exponential equation and convert it into an algebraic expression. 2. **General Formula:** Exponential equations often have the form $a^{f(x)} =

1. **State the problem:** Solve the equation $$\frac{x^2 - 3x + 2}{x - 1} = 0$$ for $x$. 2. **Recall the rule:** A fraction equals zero if and only if its numerator is zero and the

1. **State the problem:** We are given the expression $$\frac{x^2 - c}{x - b}$$ where $b$ and $c$ are positive integers. It is stated that this expression is equivalent to $$x + b$

1. **State the problem:** Simplify the expression $$\frac{x^2}{x-3} - \frac{x+15}{2x-6}$$ and express it in the form $$ax + b$$ for all $$x \neq 3$$. Find the value of $$b$$. 2. **

1. **State the problem:** We need to find the value of $k$ such that $$(4\sqrt{10})(3\sqrt{10}) = 100^k.$$\n\n2. **Rewrite the expression:** Recall that $\sqrt{10} = 10^{1/2}$. So,

1. **State the problem:** We are given that the value of $k^{-5}$ is twice the value of $4k^{-2}$. We need to find the value of $k$. 2. **Write the equation:** According to the pro

1. We are given the equation $x + xy - 2x^3 = 2$ and asked to analyze or solve it. 2. The equation involves both $x$ and $y$, so we can try to isolate $y$ in terms of $x$.

1. **بيان المسألة:** لدينا قيمة مبنى عند انتهاء البناء 4,000,000 شاقل، وهي مجموع قيمة الأرض وقيمة الشقق.

1. **State the problem:** Given the equation $4x - y = \frac{2}{3}$, find the value of $$\frac{81^{3x}}{27^y}$$. 2. **Rewrite the bases as powers of 3:**

1. **Problem statement:** Given the function $$f(x) = \frac{1}{2}x^2 e^{x+1}$$, find the domain of $$f(x)$$. 2. **Formula and rules:** The domain of a function is the set of all re

1. **Problem Statement:** An aeroplane covers a certain distance at a speed of 840 km/hr in 6 hours. We need to find the speed required to cover the same distance in $1 \frac{2}{3}

1. The problem asks to find an expression equivalent to $$\sqrt[4]{x^2 + 8x + 16}$$ where $$x > 0$$. 2. First, recognize that the expression inside the fourth root is a quadratic:

1. **State the problem:** We want to find an equivalent form of $$\sqrt[3]{54x^{5}y^{12}}$$ from the given options. 2. **Recall the cube root property:** $$\sqrt[3]{a^3} = a$$ and

1. **State the problem:** Solve the system of simultaneous equations: $$3x + y = 20$$

1. **State the problem:** Solve the system of linear equations: $$3x + y = 20$$

1. **State the problem:** Find the lowest common multiple (LCM) of 6 and 8. 2. **Formula and rules:** The LCM of two numbers is the smallest positive integer divisible by both numb

1. The problem is to graph the logarithmic function $y = \log(x)$.\n\n2. The logarithmic function $y = \log(x)$ is the inverse of the exponential function $y = 10^x$ when the base

1. **Problem Statement:** An aeroplane covers a certain distance at a speed of 840 km/hr in 6 hours. We need to find the speed required to cover the same distance in $1 \frac{2}{3}

1. সমস্যাটি হলো তিনটি রৈখিক সমসংযোজক a, b, c এবং সমীকরণ $m^2 - \frac{2m}{x} + 1 = 0$ এবং $A = \frac{2 - \sqrt{1 - y}}{2 + \sqrt{1 - y}}$ দেওয়া আছে। (ক) প্রমাণ করতে হবে $\left(\fra

1. **State the problem:** We have a pair of linear equations $5x + 7y = 1$ and $ax + by = 1$ that represent perpendicular lines. We want to find which of the given pairs of equatio

1. **Problem Statement:** Solve the equation $-3x + \frac{6}{7} = 9x^2$ and find the roots.