🧮 algebra

1. We are given the function $$f(m,n) = \prod_{k=m}^n \left(1 - \frac{1}{k}\right) = \prod_{k=m}^n \frac{k-1}{k}$$ for integers $m,n$ with $1 < m \le n$. We want to find $$S = \sum

1. **State the problem:** We have two quadratic equations with roots related as follows: - Equation 1: $2x^2 - mx + 8 = 0$ with roots $\alpha$ and $\beta$.

1. We are asked to simplify the expression $x^2 \times x$. 2. Recall the law of exponents which states that when multiplying powers with the same base, you add the exponents: $a^m

1. We are tasked with completing magic squares where each row, column, and diagonal sums to the same number. 2. For the first square:

1. **State the problem:** Evaluate the expression $$\sqrt{(98 \times 100 + 2)(100 \times 102 + 2) + (100 \times 2)^2}$$. 2. **Simplify each term:**

1. We need to evaluate the expression $$\frac{2 + \log_2 3}{1 + \log_2 3} + \frac{3 + \log_3 4}{1 + \log_3 2}$$. 2. First, let's analyze the first fraction:

1. The problem is to factorise the cubic polynomial $$x^3 + 8x^2 + 17x + 10$$. 2. We look for factors in the form $$(x + a)(x + b)(x + c)$$, where $a,b,c$ are constants.

1. **State the problem:** Given that $xy < 0$, $$\frac{1}{x^2} + \frac{1}{y^2} = 40,$$ and $$x + y = \frac{1}{3},$$ find the value of $$\frac{1}{x^4} + \frac{1}{y^4}.$$\n\n2. **Exp

1. **State the problem:** Given that \(xy < 0\), \(\frac{1}{x^2} + \frac{1}{y^2} = 40\), and \(x + y = \frac{1}{3}\), find the value of \(\frac{1}{x^4} + \frac{1}{y^4}\). 2. **Expr

1. **Stating the problem:** We are given that $xy < 0$, $$\frac{1}{x^2} + \frac{1}{y^2} = 40,$$ and $$x + y = \frac{1}{3}.$$ We need to find the value of $$\frac{1}{x^4} + \frac{1}

1. We start with the expression: $$\frac{\sqrt{50} + 7}{\sqrt{50} - 7} + \frac{\sqrt{50} - 7}{\sqrt{50} + 7}$$

1. Stating the problem: Simplify and evaluate the expression $$\frac{\sqrt{50} + 7}{\sqrt{50} - 7} + \frac{\sqrt{50} - 7}{\sqrt{50} + 7}$$. 2. To simplify, let $$a = \sqrt{50}$$ an

1. The problem is to factorise the cubic polynomial $$x^3 + 8x^2 + 17x + 10$$. 2. Since it is a cubic polynomial, we try to factor it into a product of three linear binomials: $$(x

1. **State the problem:** We need to find the value of $$\left(4 \sqrt{4} + 2 \sqrt{3} - \sqrt{49 + 8 \sqrt{3}}\right)^2$$. 2. **Simplify each term inside the parentheses:**

1. **Stating the problem:** Factor the expression $x^2$. 2. **Understanding the expression:** The expression $x^2$ means $x$ multiplied by itself, i.e., $x \times x$.

1. The problem is to find the largest prime factor of 999936. 2. Start by factorizing 999936. Notice that it is even, so divide by 2:

1. State the problem: We want to find the value of $\left(4\sqrt{4} + 2\sqrt{3} - \sqrt{49} + 8\sqrt{3}\right)^2$. 2. Simplify each square root:

1. The problem asks us to factorise the cubic polynomial $$x^3 + 8x^2 + 17x + 10$$ into linear factors. 2. To factorise, first try to find rational roots using the Rational Root Th

1. State the problem: We need to find which fraction among $\frac{5}{9}$, $\frac{4}{7}$, $\frac{3}{5}$, $\frac{6}{11}$, and $\frac{13}{21}$ has the smallest value. 2. To compare fr

1. The problem asks us to evaluate the sum of fractions: $$\frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} + \frac{1}{90} + \frac{1}{110}$$ 2. To add these

1. The problem asks us to find the value of $$\left(4\sqrt{4} + 2\sqrt{3} - \sqrt{49 + 8 \sqrt{3}} \right)^2$$. 2. Start by simplifying the terms inside the parentheses: