🧮 algebra

1. The problem asks for the definition of the floor function of a real number $x$. 2. The floor function, denoted as $\lfloor x \rfloor$, is defined as the largest integer less tha

1. **State the problem:** Factorize the expression $$4x^2 - \frac{1}{16}$$. 2. **Recognize the form:** This is a difference of squares, which follows the formula $$a^2 - b^2 = (a -

1. **State the problem:** Simplify the expression $$\frac{7a - 1}{2a^2 + 6a} + \frac{5 - 3a}{a^2 - 9}$$. 2. **Identify the denominators and factor them:**

1. **State the problem:** Simplify the expression $$\frac{7a - 1}{2a^2 + 6a} + \frac{5 - 3a}{a^7 - 8}$$. 2. **Analyze each denominator:**

1. **State the problem:** Calculate the value of $5^{16}$. 2. **Formula used:** The expression $a^n$ means multiplying the base $a$ by itself $n$ times.

1. **State the problem:** Factorize the expression $$16y^2 - \frac{36}{49}$$. 2. **Recognize the form:** This is a difference of squares because $$16y^2 = (4y)^2$$ and $$\frac{36}{

1. **State the problem:** Factorize the expression $4x^2 - 16$. 2. **Recall the formula:** This is a difference of squares, which follows the rule:

1. **State the problem:** Simplify the expression $$D(8,16) - \left[D(2,w) \cdot f7\right]^4 + \left[D(2,y)^2 \cdot D(4,8)\right]^4 + \left[D(2,y)^3 \cdot D(2,w)^2\right]^4 + \left

1. **State the problem:** We are given the polynomial $$P(x) = 3x^{279} - 2x^{170} + x^{7} - 3$$ and the divisor $$D(x) = x + 128$$. We want to evaluate $$P(-1)$$. 2. **Recall the

1. **State the problem:** We are given the system of equations: $$y - 3x = 0$$

1. Problem statement: Solve the equation $x^2 + 11 = 12$. 2. Formula and rules: To solve quadratic equations like this, isolate $x^2$ and then take the square root of both sides. R

1. **Stating the problem:** Solve the system of equations: $$y - 3x = 0$$

1. **Problem:** Rishika ranked 107th from the top and her friend ranked 54th from the bottom in a competition with 228 participants. Find how many participants ranked between them.

1. The problem asks for a possible value of $|y|$ given that $(x,y)$ is a solution to the "question above." Since no specific equation or system is provided in this message, we can

1. The problem involves simplifying and evaluating the polynomial expression $$5x^2 + xy + 3y^2$$ given values for $x$ and $y$. 2. The formula for evaluating a polynomial is to sub

1. **Problem:** Calculate $$4 \frac{1}{6} - \frac{1}{2} \times 9$$. 2. **Formula and rules:** Follow order of operations (PEMDAS): multiplication before subtraction.

1. **State the problem:** Calculate the value of the expression $3.4 - (-6.9) \times 2.4$. 2. **Recall the order of operations:** Multiplication must be done before subtraction.

1. **State the problem:** Calculate the value of $3.4 - (-6.9) \times 2.4$. 2. **Recall the order of operations:** Multiplication must be done before subtraction.

1. **State the problem:** Simplify the expression $$\frac{2x^3 + 2x^2 + 2}{(x+1)^2 (x^2 + 1)}$$. 2. **Factor the numerator:** First, factor out the greatest common factor (GCF) fro

1. **State the problem:** Solve the system of equations for $x$ and $y$: $$\frac{x}{6} - \frac{y}{2} = \frac{1}{6}$$

1. **State the problem:** Solve the system of equations: $$\frac{2}{5}X - Y = 4$$