🧮 algebra

1. We are asked to simplify the expression $$18^{-\frac{1}{2}} \cdot 2^{-\frac{1}{2}}$$. 2. Recall the rule for exponents: $$a^{-b} = \frac{1}{a^b}$$ and the product rule: $$a^m \c

1. **State the problem:** Evaluate the expression $$\left(\frac{1}{4} + \frac{2}{9}\right) \cdot \frac{6}{7}$$ and write the answer in simplest form. 2. **Add the fractions inside

1. **State the problem:** Simplify the expression $$\frac{(16)^{-2}}{(4^{-1})^2}$$. 2. **Recall the exponent rules:**

1. Vamos a resolver el primer problema: $2\pi x = -3 \frac{1}{3} \sqrt{25 - \sqrt{7}} - 1.4$. 2. Primero, recordemos que $\pi$ es aproximadamente $3.1416$ y que $-3 \frac{1}{3}$ es

1. El primer ejercicio es la expresión algebraica: $g + xg^ + \frac{7x8}{g}$. Parece que hay un error tipográfico en $xg^$, asumiendo que es $xg$. Entonces la expresión es $g + xg

1. **State the problem:** Find the geometric mean between 5 and 10. 2. **Formula:** The geometric mean $G$ between two numbers $a$ and $b$ is given by:

1. **State the problem:** Simplify the expression $3 \log 10 + 2 \log 10$. 2. **Recall the logarithm property:** For any logarithm, $a \log b = \log b^a$.

1. Planteamos el problema: Resolver el sistema de ecuaciones por el método de sustitución: $$\begin{cases} x - y = 1 \\ x + y = 1 \end{cases}$$

1. **State the problem:** Solve the equation $$\frac{xe^x - 3e^x}{5x^2 - 20x + 20} = 0$$ for $x$. 2. **Understand the zero fraction rule:** A fraction equals zero if and only if it

1. The problem is to determine whether the correct term is "lne" or "1ne". 2. This is a question about notation, not a mathematical operation.

1. The problem is to evaluate $\ln e^{2}$. 2. Recall the logarithm power rule: $\ln a^{b} = b \ln a$.

1. **State the problem:** Simplify $$\left(4x^{-3}y^{5}\right)^3 \times \left(125x^{-6}y^{12}\right)^{\frac{2}{3}}$$. 2. **Apply the power of a product rule:** $$(ab)^m = a^m b^m$$

1. Let's state the problem: How to calculate the final price of an item after applying a discount and then adding sales tax. 2. The formula to find the final price is:

1. **State the problem:** Solve the equation $$\frac{1}{2}[2(x+1) -(x-3)^2] = \frac{1}{3}[3(x-1) -(x-3)^2].$$ 2. **Rewrite the equation:**

1. **Problem statement:** Prove the summation properties: $$\sum_{k=1}^n (x_k + y_k) = \sum_{k=1}^n x_k + \sum_{k=1}^n y_k$$

1. **State the problem:** Simplify the expression $$\sqrt{x^5} \times \sqrt{x}$$. 2. **Recall the property of square roots:** For any non-negative $a$ and $b$, $$\sqrt{a} \times \s

1. **State the problem:** Calculate the value of the expression $\frac{75 \cdot 175^3}{12}$.\n\n2. **Recall the order of operations:** We first calculate the exponent, then multipl

1. **State the problem:** Calculate the value of the expression $\frac{75 \cdot 175^2}{6}$. 2. **Recall the order of operations:** First, calculate the exponent, then multiply, and

1. The problem is to describe the solution set for the inequality $x \leq -2$ or $x > 5$ on a number line ranging from -10 to 10. 2. The inequality $x \leq -2$ means all values of

1. **State the problem:** Solve the system of linear equations: $$\begin{cases} x - 3y + 3z = -10 \\ x - 2y + 4z = -3 \\ x - 2y + 2z = -7 \end{cases}$$

1. The problem is to find the solution to the system of equations given by the two lines: Line A: $y=\frac{1}{2}x + 8$