🧮 algebra

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

1. **State the problem:** Factorise the quadratic expression $x^2 + 18x + 17$. 2. **Recall the formula:** To factorise a quadratic $ax^2 + bx + c$, we look for two numbers that mul

1. **Problem Statement:** We are given the function $$y = x(x - 4) - 5$$ which simplifies to $$y = x^2 - 4x - 5$$.

1. The problem is to determine if a function is surjective (onto). 2. A function $f: A \to B$ is surjective if for every element $b$ in the codomain $B$, there exists at least one

1. The problem is to determine if a function $f$ is injective (one-to-one). 2. A function $f$ is injective if and only if for every $x_1$ and $x_2$ in the domain, whenever $f(x_1)

1. **Problem 1: Find the value of 5c given the mapping:** Given the mapping:

1. **State the problem:** A man takes 60 minutes less than his apprentice to paint a room. Together, they take 72 minutes to paint the room. We need to find the time the apprentice

1. **State the problem:** Given the mapping from $x$ to $y$ values: $$\begin{array}{c|cccccc}

1. **State the problem:** A man takes 60 minutes less than his apprentice to paint a room. Together, they take 72 minutes to paint the room. We need to find the time the apprentice

1. **State the problem:** Solve the system of equations using the method of substitution: $$x - 2y = 3$$