🧮 algebra

1. **State the problem:** Simplify the expression $$\frac{\sqrt{32b^{10} c^6}}{\sqrt{9b^{12} c^4}}$$ assuming all variables represent positive numbers. 2. **Recall the property of

1. **Stating the problem:** A pool is being filled with water, and every minute, 8 additional gallons of water are added. We want to find the total amount of water in the pool afte

1. **Stating the problem:** Simplify the expression $$(-2a)^3 (-3a)(-b^2)^3.$$ 2. **Recall the rules:**

1. **Problem statement:** Given functions $h(x) = x + 1$, $g(x) = x^2$, and $f(x) = x - 2$, find various compositions of these functions. 2. **Recall function composition:** For fu

1. **Stating the problem:** Calculate and simplify the expressions for problems 393, 394, and 395.

1. **Énoncé du problème** : Trouver la dérivée de la fonction $$f(x) = \frac{5}{4x^2 + 2}$$ sur son domaine de définition $$D = \mathbb{R}$$. 2. **Formule utilisée** : Pour dériver

1. **Stating the problem:** We have a sequence of dot patterns where each pattern adds the same number of dots compared to the previous one. We want to find an expression for the n

1. **State the problem:** Solve the equation $5x + 2 = 2x + 11$ for $x$. 2. **Write down the formula and rules:** To solve linear equations, we isolate the variable $x$ by performi

1. The problem asks to find the number of tiles in the 8th pattern of a sequence where the number of tiles in the $n^{th}$ pattern is given by the formula: $$\text{Tiles} = 4n + 1$

1. **State the problem:** We need to evaluate the equation for the Orange row, which is $$-y - w + 1 = -2$$, given that $y = -5$ and $w = -2$. 2. **Substitute the values:** Replace

1. **State the problem:** We need to find the value of the expression $$v + w + x + y + z = 8$$ given the values of the variables.

1. The problem is to solve the equation graded to 6 levels, which typically means solving a polynomial or equation of degree 6. 2. The general approach to solving a polynomial equa

1. Stating the problem: Solve the equation $$3x + 32.1 = 5x - 0.3$$. 2. Write down the equation:

1. We start with the function $f(x) = x + \sqrt{x^{2} - 4}$. We want to find its inverse function $f^{-1}(x)$. 2. To find the inverse, we set $y = x + \sqrt{x^{2} - 4}$ and solve f

1. **State the problem:** Simplify the expression $\sqrt{45x^{19}}$ assuming $x>0$. 2. **Recall the rule:** $\sqrt{a b} = \sqrt{a} \times \sqrt{b}$ and $\sqrt{x^{2n}} = x^n$ for $x

1. The problem is to understand the mathematical constant $e$. 2. The number $e$ is an irrational constant approximately equal to 2.71828.

1. The problem is to evaluate the function $$\tau(30) = 0.9 - e^{-0.2 \times 30}$$ and understand its behavior. 2. The formula involves an exponential decay term $$e^{-kt}$$ where

1. **State the problem:** Write the expression $\sqrt{75x^{20}}$ in radical form assuming $x>0$. 2. **Recall the formula:** The square root of a product is the product of the squar

1. **State the problem:** Find the x- and y-intercepts of the line given by the equation $$3x - 4y = 12$$ and use them to sketch the line. 2. **Recall intercept definitions:**

1. **Problem (a):** Tom's car has lost 18% of its value and is now worth 18450. 2. **Formula:** If the original value is $V$, then after losing 18%, the value is $V - 0.18V = 0.82V

1. **State the problem:** Solve the system of linear equations: $$\begin{cases} 2x - 3y = 2 \\ x + 3y = 10 \end{cases}$$