🧮 algebra

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 $x$.

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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.

1. **Stating the problem:** We need to find the number replacing the question mark (?) inside the third triangle given the pattern from the first two triangles. 2. **Observing the

1. **State the problem:** Simplify or analyze the expression $x^3 \ln \sqrt{x^2+1}$. 2. **Recall relevant formulas and rules:**

1. **State the problem:** Solve the recurrence relation $$a_n - 5a_{n-1} + 6a_{n-2} = 2^n, \quad n \geq 2$$ with initial conditions $$a_0 = 1, \quad a_1 = 1$$ using generating func

1. The problem involves understanding the relationship between two variables: $t$ (the number of tickets Patrick purchases) and $r$ (the number of rides Patrick can go on). 2. We a

1. The problem asks to identify the independent variable given two variables: $c$ (number of costumes) and $d$ (number of dances). 2. By definition, the independent variable is the

1. The problem involves understanding the relationship between two variables: $b$ (the number of bags Johnny brings to the store) and $p$ (the number of products on Johnny's shoppi