🧮 algebra

1. **State the problem:** Simplify the expression $$\frac{x-4}{7} \cdot \frac{x}{21}$$. 2. **Multiply the numerators and denominators:**

1. We are asked to express the sum $5 + \frac{1}{y}$ as a single fraction. 2. To combine these terms, we need a common denominator. The denominator here is $y$.

1. **Calcule a característica da matriz A.** A matriz A é dada por:

1. **State the problem:** Solve the equation $$4^x \cdot 16^{x+1} = \left( \frac{1}{2^{-5}} \right)^2$$ for $x$. 2. **Rewrite bases as powers of 2:**

1. **Problem 9:** Simplify $$\frac{(10^3 \cdot 20^2)^3}{5^5 \cdot 2} \cdot \frac{250^4 \cdot 8^3}{75^2}$$ 2. First, simplify inside the parentheses:

1. The problem asks to find the domain of two functions, labeled as question a and question b. 2. To find the domain of a function, identify all values of $x$ for which the functio

1. The problem asks for the domain of variables $a$ and $b$. 2. The domain of a variable is the set of all possible values it can take.

1. The problem is to simplify the expression $$\frac{1}{\sqrt{3} - 1}$$. 2. To simplify, multiply numerator and denominator by the conjugate of the denominator $$\sqrt{3} + 1$$ to

1. Problem a: Graph the function $f(x) = 5x^4 - 13$ on the window $[-2,2] \times [-16,16]$ and state its domain and range. 2. The domain of $f(x)$ is all $x$ values in the interval

1. The problem asks to fill in the blanks about functions and answer questions about independent and dependent variables. 2. A function is a rule that assigns to each value of the

1. **State the problem:** Simplify the expression $4y^3 + 16y$. 2. **Factor out the greatest common factor (GCF):** Both terms have a factor of $4y$.

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 $4$.