🧮 algebra

1. **State the problem:** Find the value of $\left(2^{-5} \div 2^{-2}\right)^2$. 2. **Recall the rule for division of exponents with the same base:**

1. **State the problem:** We have a volume function for a box given by $$f(x) = (30 - 2x)(50 - 2x)(x)$$ and need to calculate the volume at $x=10$ cm. 2. **Calculate $f(10)$:** Sub

1. مسئله را بیان می‌کنیم: فاصله مستقیم بین دو شهر A و B برابر 240 کیلومتر است. راننده در مسیر رفت با سرعتی 20 کیلومتر بر ساعت بیشتر از مسیر برگشت حرکت می‌کند، اما در مسیر رفت 30 دق

1. State the problem. Problem: Find the domain and range of the function $f(x)=x^2+6x+5$.

1. **State the problem:** We need to solve the equation $2x + 3 = 11$ for $x$. 2. **Formula and rules:** To solve a linear equation, isolate the variable on one side by performing

1. **State the problem:** Solve the linear equation $3x + 5 = 20$ for $x$. 2. **Formula and rules:** To solve for $x$, isolate the variable by performing inverse operations. Subtra

1. **State the problem:** We start with an equal number of predator and prey mites. After one week, the prey population increases by 2700% and the predator population increases by

1. **State the problem:** We have two functions:

1. **Problem Statement:** Graph the function and analyze its behavior using the Leading Term Test, find zeros and their multiplicities, identify additional points, and sketch the g

1. **State the problem:** We need to find digits $x$ and $y$ such that the number $42x4y$ is divisible by 72. 2. **Understand divisibility rules:** A number is divisible by 72 if a

1. **State the problem:** Solve for $x$ in the equation $$\frac{2x + 5}{3} - \frac{x - 7}{4} = \frac{3x - 2}{6} + 1.$$\n\n2. **Identify the formula and rules:** To solve equations

1. Problem statement. The shop currently sells 25000 pairs of shoes annually.

1. **Problem Statement:** Determine the domain and range of the function $$f(x) = \frac{1}{x^2}$$. 2. **Domain:** The domain of a function is the set of all possible input values (

1. The problem asks for the coefficient of $x^2 y^3$ in the expansion of $(x + y)^5$. 2. We use the Binomial Theorem, which states:

1. The problem is to simplify the expression $$\frac{7}{2} + \frac{7}{3}$$. 2. To add fractions, we need a common denominator. The denominators here are 2 and 3.

1. **Problem Statement:** Find the coefficient of $x^2 y^3$ in the expansion of $(x + y)^5$. 2. **Formula Used:** The binomial expansion of $(x + y)^n$ is given by:

1. **Problem statement:** Given functions $f(x) = x + 2$ and $g(x) = x^2 - 9$, find $g(f(x))$. 2. **Formula and rules:** To find $g(f(x))$, we substitute $f(x)$ into $g(x)$, i.e.,

1. The problem asks for the coefficient of $x^3$ in the expansion of $(1 + x)^5$. 2. We use the Binomial Theorem formula for expansion: $$(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n

1. مسئله: تعداد جواب‌های معادله $$\sqrt{x-1} + \sqrt{x-4} = \sqrt{x-5}$$ را بیابید. 2. دامنه تعریف: برای رادیکال‌ها باید زیر رادیکال‌ها غیرمنفی باشند:

1. **State the problem:** We are given a function $h(t)$ defined by the table: | t | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

1. The problem asks for the sum of the infinite geometric series: $$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$$ 2. The formula for the sum $S$ of an infinite geometric se