🧮 algebra

1. The problem states: Simplify the expression $x^0 \times y^0$ and check if it equals $xy$. 2. Recall the rule: Any nonzero number raised to the power of zero equals 1. That is, f

1. **Planteamiento del problema:** Resolver la ecuación logarítmica $\log x + \log 4 = 0$. 2. **Fórmula y reglas importantes:**

1.1 Consider the pattern: 10; 7; 4; 1; ... 1.1.1 Determine the formula $T_n$, the general term of the sequence.

1. Consider the pattern: 10; 7; 4; 1; ... 1.1 Determine the formula $T_n$, the general term of the sequence.

1. **State the problem:** Multiply $x^{-2}$ by $x^{-3}$. 2. **Recall the rule for multiplying powers with the same base:**

1. **State the problem:** Simplify or evaluate the expression $2^N + 2$ for a given value of $N$. 2. **Formula and rules:** The expression consists of an exponential term $2^N$ plu

1. **State the problem:** Find the equation of the straight line passing through the points $(0,7)$ and $(2,0)$ in the form $y = mx + c$. 2. **Formula used:** The equation of a lin

1. The problem is to find the 4 roots of the equation for each integer value of $k$ from 0 to 10. 2. Since the original equation is not provided, let's assume a general quartic equ

1. **State the problem:** We need to find the equation of a line passing through point $P(0,0)$ that is perpendicular to the line given by $$y = -9x - 1.$$\n\n2. **Recall the slope

1. The problem is to simplify the expression $10^{-4}$. 2. Recall the rule for negative exponents: $a^{-n} = \frac{1}{a^n}$ where $a \neq 0$.

1. **State the problem:** We need to find the roots of the polynomial $$s^4 + 2s^3 + (3+K)s^2 + (1+K)s + (1+K)$$ for real values of $K$ in the range $0 \leq K \leq 5$. 2. **Polynom

1. The problem is to find the roots of a polynomial equation where $K$ is a real number. 2. Since the exact polynomial is not provided, the roots depend on the specific equation in

1. Let's start by stating the problem: Is the binomial theorem related to mathematical induction in any way? 2. The binomial theorem states that for any positive integer $n$ and an

1. **State the problem:** We want to prove by mathematical induction that the expression $$P_n = n(n^2 + 5)$$ is divisible by 6 for all integers $$n \geq 1$$.

1. **State the problem:** We need to find the roots of the polynomial $$s^4 + 2s^3 + (3+K)s^2 + (1+K)s + (1+K)$$ for values of $K$ in the interval $0 < K \leq 5$. 2. **Polynomial a

1. **State the problem:** We need to find the missing numbers in sequences 8, 9, and 10. 2. **Sequence 8:** 120, 100, 80, __, 40.

1. **State the problem:** Explain the use and order of greater than and less than symbols in inequalities. 2. **Symbols:** The less than symbol is $<$ and means "smaller than." The

1. **State the problem:** We need to evaluate the matrix product and simplify the expression:

1. **State the problem:** We are given the function $y = x(x-4) - 5$ and asked to draw its graph for $-3 \leq x \leq 4$.

1. **Problem:** Find the range of the function $$f(x) = \left| x - \frac{1}{2} \right| - 2$$. 2. **Formula and rules:** The absolute value function $$|x|$$ is always non-negative,

1. **State the problem:** Simplify the expression $ (2 - x)^2 $. 2. **Recall the formula:** The square of a binomial $ (a - b)^2 $ is given by the formula $$ (a - b)^2 = a^2 - 2ab