🧮 algebra

1. **State the problem:** Simplify the expression $$\frac{18^3 \cdot 12^2 \cdot 128}{6^6 \cdot 24^{-5}}$$. 2. **Rewrite bases in prime factorization:**

1. **State the problem:** We need to find the maximum number of students that can go on a trip with the condition that there is at least one adult for every seven students, and the

1. **Solve the inequality** $\frac{x^2 - 9x + 20}{x - 6} \leq 0$. Factor the numerator: $x^2 - 9x + 20 = (x - 4)(x - 5)$.

1. **State the problem:** Simplify the expression $$\frac{(14^3 \cdot 2^4) \cdot 7^2}{28^5}$$. 2. **Rewrite the bases in terms of prime factors:**

1. Problem 1: Solve the simultaneous equations: $$\begin{cases} x^2 + 9xy + y^2 = 23 \\ xy + x + y = 5 \end{cases}$$

1. The problem is to solve the equation $\ln x = -2$ for $x$. 2. Recall that the natural logarithm function $\ln x$ is the inverse of the exponential function $e^x$.

1. The problem is to understand how to express $x$ as $e$ raised to the power of $-2$. 2. The expression $x = e^{-2}$ means that $x$ is equal to the exponential function with base

1. **State the problem:** Simplify the expression $$\frac{14^{2} \cdot 7^{3} \cdot 98^{5} \cdot 18^{5} \cdot 63}{49^{3} \cdot 81^{4} \cdot 28^{3}}$$

1. **State the problem:** We need to analyze the quadratic expression $x^2 - x - 30$. 2. **Factor the quadratic:** To factor $x^2 - x - 30$, find two numbers that multiply to $-30$

1. The problem is to factor the quadratic expression $x^2 + 4x - 21$. 2. To factor, we look for two numbers that multiply to $-21$ and add to $4$.

1. The problem is to analyze the quadratic expression $x^2 - 12x + 35$. 2. First, we can factor the quadratic to find its roots. We look for two numbers that multiply to $35$ and a

1. **State the problem:** Simplify the expression $5 - 3(g - 2)$. 2. **Distribute the $-3$ across the parentheses:**

1. **State the problem:** Factor the quadratic expression $x^2 + 15x + 54$. 2. **Identify coefficients:** The quadratic is in the form $ax^2 + bx + c$ where $a=1$, $b=15$, and $c=5

1. The problem is to multiply $2.7 \times 10^{-4}$ by $6.3 \times 10^{6}$. 2. Multiply the decimal parts: $2.7 \times 6.3 = 17.01$.

1. The problem is to evaluate the expression $\frac{36}{2} + 2 \times 2$. 2. According to the order of operations (PEMDAS/BODMAS), we first perform division and multiplication from

1. The problem asks if the equation $6(5-2)=18$ is true. 2. First, simplify inside the parentheses: $5-2=3$.

1. The problem is to factor the expression $m^2 - 36n^2$. 2. Recognize that this is a difference of squares, which has the general form $a^2 - b^2 = (a - b)(a + b)$.

1. The problem asks to sketch a graph with the following characteristics: - Domain: $[-2, 3]$

1. Solve the equation $3x - 8 = \frac{3x}{2} + 10$. Step 1: Multiply both sides by 2 to clear the fraction:

1. **State the problem:** Leo is building a rectangular garden bed. Each side requires $2 \frac{3}{4}$ meters of wood per layer, and he plans $1 \frac{2}{3}$ layers in height. Each

1. The problem is to analyze the expression $3x^2 + 6$. 2. This is a quadratic expression where the coefficient of $x^2$ is 3 and the constant term is 6.