🧮 algebra

1. The problem defines a binary operation $*$ on real numbers such that for any real numbers $a$ and $b$, $$a * b = \frac{a}{b} + \frac{b}{a}$$

1. **Stating the problem:** Simplify the expressions: - $5 \times 3 = \frac{490}{630} + \frac{2}{5} \times \frac{3}{4}$

1. The problem is to simplify the expression $(2\sqrt{3}-3)(2\sqrt{3}+3)$.\n\n2. Recognize this as a product of conjugates of the form $(a-b)(a+b) = a^2 - b^2$. Here, $a = 2\sqrt{3

1. **Stating the problem:** We want to understand how logarithms with any base $b$ can be expressed using the natural logarithm base $e$. 2. **Formula:** The change of base formula

1. The problem is to understand and represent the inequalities on a number line: $x < 2$, $2 < x < 3$, $3 < x < 4$, and $x > 4$. 2. These inequalities divide the number line into f

1. Stating the problem: Simplify the expression $$(2\sqrt{3}-3)(2\sqrt{3}+3)$$. 2. Recognize this as a difference of squares pattern: $$(a-b)(a+b) = a^2 - b^2$$ where $a = 2\sqrt{3

1. The problem involves evaluating inequalities and expressions given as: - $x7 - 4$

1. Solve for $x$: 1.1.1 Solve $x^2 - 5x = 0$:

1. The problem is to simplify the expression $4x^2 + 12xy$. 2. Identify the common factors in both terms. Both terms have a factor of $4x$.

1. Let's start by understanding the number $e = 2.7182818$. 2. The number $e$ is the base of the natural logarithm, which means the logarithm with base $e$ is called the natural lo

1. **Stating the problem:** We need to solve the equation $$\frac{x^2}{3x + 5} = \frac{1}{2}$$ for the possible values of $x$. 2. **Cross-multiply to eliminate the fractions:**

1. The problem is to simplify or understand the expression $$\frac{-8}{x-2}$$. 2. This expression represents a rational function where the numerator is a constant $-8$ and the deno

1. **State the problem:** We have a binary operation defined as $a*b = \frac{a}{b} + \frac{b}{a}$ for real numbers $a$ and $b$. 2. We are given the equation $((\sqrt{x})+1)*((\sqrt

1. The problem involves identifying and continuing patterns in sequences of shapes and numbers. 2. For the numeric sequences:

1. **State the problem:** We have a binary operation $*$ defined on real numbers by the rule $a * b = \frac{a}{b} + \frac{b}{a}$. We want to understand this operation and possibly

1. The problem asks us to find a rational number between $5.01$ and $5.04$. 2. A rational number is any number that can be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ a

1. The problem asks which numbers are greater than $-0.33$. We will compare each choice to $-0.33$. 2. Choice A: $-0.3$. Since $-0.3$ is greater than $-0.33$ (because $-0.3$ is clo

1. The problem asks which numbers are greater than $$-\frac{185}{100}$$. 2. First, convert $$-\frac{185}{100}$$ to decimal: $$-\frac{185}{100} = -1.85$$.

1. The problem is to calculate $8 \times 6 \frac{1}{4}$ using the distributive law. 2. First, convert the mixed number $6 \frac{1}{4}$ to an improper fraction: $$6 \frac{1}{4} = \f

1. The problem asks to match the given numbers with the correct labels a, b, and c on the number line. 2. The points on the number line are:

1. **State the problem:** Factorise fully the expressions $2(x + 5)$ and $50 - 2y^2$. 2. **Factorise $2(x + 5)$:**