🧮 algebra

1. **State the problem:** Simplify the expression $$\left(\frac{16}{y^8}\right)^{\frac{1}{4}} \times y^{-3}$$. 2. **Recall the rules:**

1. **State the problem:** Solve the quadratic equation $$x^2 + 4x - 30 = 0$$. 2. **Recall the quadratic formula:** For any quadratic equation $$ax^2 + bx + c = 0$$, the solutions a

1. **State the problem:** We need to find the coordinates $(x,y)$ such that the distance from the point $(x,y)$ to the point $(-1,5)$ is $\sqrt{2}$. This is given by the equation $

1. **State the problem:** We need to find the interest rate $r$ given the equation $5066.55 = 4500(1 + \frac{r}{100})^5$. 2. **Formula used:** This is a compound interest formula:

1. مسئله را بیان می‌کنیم: معادله داده شده به صورت $$y = C_1 e^x + C_r e^{-x} + C_u x_r e^x$$ است. 2. این معادله ترکیبی از توابع نمایی است که شامل $$e^x$$، $$e^{-x}$$ و $$x_r e^x$$

1. مسئله را بیان می‌کنیم: معادله داده شده به صورت $$y = C_1 e^x + C_2 e^{-x} + C_3 x^2 e^x$$ است. 2. این معادله ترکیبی از توابع نمایی و چندجمله‌ای است که شامل سه جمله با ضرایب ثابت

1. **State the problem:** Determine whether the function $f(x) = \frac{x^2}{x^2 + 4}$ is even, odd, or neither. 2. **Recall definitions:**

1. The problem describes a parallelogram with sides labeled by algebraic expressions and inequalities relating variables. 2. Recall that in a parallelogram, opposite sides are equa

1. Evaluate the following logarithms: 1.a) Given $\log_{17} 289$, recognize that $289 = 17^2$. Using the logarithm rule $\log_b b^k = k$, we get:

1. समस्या: हमें अभिव्यक्ति $ab \times AA$ का मान निकालना है। 2. सबसे पहले, यह समझना जरूरी है कि $ab$ और $AA$ क्या हैं। यदि ये दो अंकों वाले संख्याएँ हैं, तो $ab$ का मतलब है $10a +

1. Problem statement: Find the complex conjugate of a complex number $A$. 2. Formula: The complex conjugate of a complex number $A = a + bi$ (where $a$ and $b$ are real numbers and

1. **State the problem:** Determine whether the function $f(x) = \frac{x^2}{x^2 + 4}$ is even, odd, or neither. 2. **Recall definitions:**

1. **State the problem:** Solve the equation $$9 = 2y - y^2 - 6x - x^2$$ for $x$ and $y$ or analyze its form. 2. **Rewrite the equation:** Move all terms to one side to get a stand

1. **State the problem:** We are given the equation $$16 + x^2 + y^2 - 8x - 6y = 0$$ and need to identify the geometric shape it represents and find its center and radius if it is

1. **State the problem:** Solve the equation $$x^2 + 3x + 4\sqrt{x^2 + 3x - 6} = 18$$ for $x$. 2. **Introduce substitution:** Let $$y = \sqrt{x^2 + 3x - 6}$$. Then $$y^2 = x^2 + 3x

1. **State the problem:** We are given the equation $y^2 + 4x - 20 - 2y = -x^2$ and asked to analyze it as a circle.

1. **Problem Statement:** A computer training course lasts 15 days. Practical training starts at 75 minutes on day 1 and increases by 15 minutes each subsequent day.

1. **State the problem:** Simplify the expression $2(3x-2)(3x-2)$. 2. **Formula and rules:** When multiplying expressions like $(a-b)(a-b)$, use the formula for the square of a bin

1. **Problem Statement:** We are given a budget constraint and minimum purchase requirements for a pizza party.

1. Problem statement: Prove that $\sqrt[3]{3}$ is irrational. 2. Formula and concept: A number is irrational if it cannot be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$

1. **Problem Statement:** Find the Highest Common Factor (H.C.F) of 144 and 96. 2. **Formula and Concept:** The H.C.F (also called GCD) of two numbers is the largest number that di