🧮 algebra

1. Planteamos el problema: escribir $\sqrt{2} \cdot \sqrt[3]{2}$ como un solo radical y simplificar. 2. Recordamos que $\sqrt{2} = 2^{\frac{1}{2}}$ y $\sqrt[3]{2} = 2^{\frac{1}{3}}

1. Planteamos el problema: Simplificar la expresión $$\left[\frac{\left(3(a+b)^{-2} c\right)^3 \left[2(a+b) c^3\right]^{-1}}{12 (a+b)^3 c^{-1}}\right]^5 \quad \text{con } c \neq 0,

1. Primero, escribamos la expresión original: $$\left[\frac{(x+y)^{-2} \cdot 3x^{2} \cdot (x+y)^{4} \cdot y^{2}}{x^{-2} \cdot y^{3}}\right]^3$$

1. The problem is to solve the quadratic equation $x^2 - 5x + 6 = 0$ using a different method from factorization, for example, the quadratic formula. 2. Recall the quadratic formul

1. Planteamos el problema: Simplificar $$\left( \frac{4 (a+b)^{-4} a^{2} b^{3}}{8 (a+b)^{3} a^{-2} b} \right)^2$$ con las condiciones $$a \neq 0, b \neq 0, a+b \neq 0$$. 2. Simplif

1. The problem asks to show that when $x$ is very small such that terms involving $x^4$ and higher powers can be neglected, $$\frac{e^x}{1+x} = 1 + \frac{x^2}{2} - \frac{x^3}{3}.$$

1. Planteamos el problema: Simplificar la expresión $$\left( a^{-1} + b^{-1} \right)^{-1}$$ con $a \neq 0$, $b \neq 0$, y $a + b \neq 0$. 2. Primero, recordemos que $a^{-1} = \frac

1. The request is to analyze a graph, but no graph data or description was provided.\n2. To assist effectively, please provide the function, equation, or graph details.\n3. Once th

1. The problem is to simplify the expression $y^2$ for small values of $\theta$ using Taylor expansions. 2. Recall the Taylor expansion of cosine near zero:

1. The problem is to solve the equation $$\frac{t}{t+16} + \frac{t+16}{t} = \frac{2 \times 1}{12} = \frac{1}{6}.$$\n\n2. Let's rewrite the equation clearly:\n$$\frac{t}{t+16} + \fr

1. The user asked about "equation 4" but did not provide the equation or context. 2. To help, please provide the full equation or problem statement involving "equation 4".

1. **State the problem:** Write the intervals describing the given inequalities using interval notation. 2. **For i) $x \succ 3$: **This means all real numbers greater than 3, not

1. Problem: Simplify the fraction $\frac{5}{3}$. 2. Explanation: The fraction $\frac{5}{3}$ is already in its simplest form because the numerator 5 and denominator 3 have no common

1. **Problem statement:** The marked price of an electric iron is 780. The shopkeeper allows a discount of 10% and still gains 8%. We need to find the gain percent if no discount i

1. Solve the equation $25x^4 - 104x^2 + 16 = 0$. 2. Start by using substitution $u = x^2$ to simplify the quartic to a quadratic in $u$:

1. **Stating the problem:** Simplify the expression $$(\sqrt{2} - \sqrt{3})(\sqrt{2} + \sqrt{3}) - 2(\sqrt{2} + \sqrt{3})$$. 2. **Simplify the first product using the difference of

1. **Problem:** Simplify the expression $$\sqrt{2 - \sqrt{3}}(\sqrt{2} + \sqrt{3}) - 2(\sqrt{2} + \sqrt{3})$$. 2. **Step 1:** Let $$a = \sqrt{2 - \sqrt{3}}$$ and $$b = \sqrt{2} + \

1. The problem is to factor the polynomial equation $$2x^4 - 7x^2 - 4 = 0$$. 2. Notice the equation is quadratic in form if you let $$y = x^2$$, so rewrite it as $$2y^2 - 7y - 4 =

1. State the problem: Solve the quadratic equation $$x^2 - 144 = 0$$. 2. Move constant term to the other side: $$x^2 = 144$$.

1. The problem is to find the square root (racine) of 25. 2. By definition, the square root of a number $x$ is a number $y$ such that $y^2 = x$.

1. Stating the problem: Find $q$ given by the formula $q = \sqrt[3]{2r} - r^2$ where $r$ is a variable. 2. Understand the expression: The first term is the cube root of $2r$, writt