🧮 algebra

1. **State the problem:** Calculate the value of $$\sqrt{\left(2 \frac{3}{5}\right)^2 - \left(2 \frac{2}{5}\right)^2}$$.

1. The problem asks to identify which graph corresponds to the function $f(x) = e^x$. 2. The function $f(x) = e^x$ is an exponential growth function with base $e \approx 2.718$.

1. **Problem statement:** We have a quadratic function $f(x) = 2x^2 - 6x + 3$ and a point $V = \left(\frac{3}{2}, -\left(\frac{3}{2}\right)^2\right)$ claimed to be the vertex of th

1. **State the problem:** We are given a quadratic function $f(x) = ax^2 + bx + c$ with $a > 0$ and asked if $f$ is bounded from above. 2. **Recall the shape of the parabola:** Sin

1. **State the problem:** We are given the function $u = e^{\frac{x}{y}} + e^{\frac{y}{x}}$ and want to understand or analyze it. 2. **Recall the exponential function:** The expone

1. The problem is to evaluate the summation $$\sum_{i=4}^5 i$$. 2. The summation notation $$\sum_{i=a}^b f(i)$$ means to add the values of the function $$f(i)$$ for each integer $$

1. **Problem Statement:** Find the pattern (rule) of the sequence: $\frac{4}{6}, \frac{12}{7}, 4, 2, -12, -\frac{28}{5}, -4, \ldots$ 2. **Step 1: Simplify fractions where possible.

1. **Stating the problem:** You provided matching problems for laws of exponents and simplification exercises. We will verify your answers for correctness. 2. **Matching Column A t

1. **State the problem:** We need to find a combination of the given numbers that add up to 6224.35. 2. **List the numbers:** 69.40, 4440.51, 208.27, 359.85, 2591.57, 360.00, 180.0

1. **Problem Statement:** We want to understand why for the function $$y = \frac{x^2 - 3x + 4}{x^2 + 3x + 4}$$ with real $x$, the discriminant condition $D \geq 0$ is used and why

1. The problem is to solve the equation $$\frac{x}{8!} + \frac{x}{9!} = \frac{1}{10!}$$ for $x$. 2. Recall the factorial values and the property that $n! = n \times (n-1)!$. Here,

1. **Énoncé du problème :** Encadrer les expressions $$\frac{x+y}{x-y}$$ et $$x^2 - y^2$$ pour $$-\frac{3}{2} \leq x \leq -\frac{1}{2}$$ et $$\frac{5}{2} \leq y \leq \frac{7}{2}$$.

1. **State the problem:** We have a list of numbers and want to proportion them according to percentages 35%, 15%, and 50%. 2. **Understand the percentages:** The percentages 35%,

1. Polinomial adalah ekspresi matematika yang terdiri dari variabel dan koefisien yang dihubungkan dengan operasi penjumlahan, pengurangan, dan perkalian. 2. Contoh polinomial: $$3

1. The problem is to find the values after 0, 1, 2, 3, 4, and 5 years for a given sequence or function. 2. Since the user did not specify the function or rule governing the sequenc

1. We are given the formula for the tiger population after $t$ years: $$a = 4000 + 50 \sqrt{t}$$ 2. The problem asks to find the population after 4 years, so we substitute $t=4$ in

1. **Problem statement:** Given functions $f(x) = -3x^2 + 1$ and $g(x) = 2^x - 4$, solve the following. 2. **Step 2.1: Determine $f(-3)$**

1. **State the problem:** We know 8 machines can assemble 6 printers in 10 hours. We want to find how many hours 5 machines will take to assemble 12 printers.

1. **Problem:** Calculate 22 (2/9) % of 1080. 2. **Formula:** To find a percentage of a number, use the formula:

1. Problema: Găsiți numărul natural $n$ pentru care $$8^n + 8^{n+1} = 36 \cdot 2^{2011}.$$ 2. Formula și reguli importante:

1. **Stating the problem:** We are given a graph of a function and asked to find its three roots. Roots are the values of $x$ where the function crosses the $x$-axis, i.e., where $