🧮 algebra

1. Simplify \(\left(3x^2 y^3\right)^2\): Step 1: Apply the power to each factor inside the parentheses:

1. **State the problem:** Given $a=1194$ and $b=945$, verify and explore properties of $\gcd(a,b)$ and $\mathrm{lcm}(a,b)$, simplify expressions involving powers of $a$ and $b$, an

1. Calculons les expressions demandées : - $\sqrt{49} - 2\sqrt{36} = 7 - 2 \times 6 = 7 - 12 = -5$

1. Problema: Risolvere la disequazione $\frac{2x-1}{3} > \frac{x-4}{2} + 1$. Passo 1: Portiamo tutto al denominatore comune o moltiplichiamo entrambi i membri per 6 (il minimo comu

1. **Stating the problem:** You asked which numbers can be used in synthetic division. 2. **Explanation:** Synthetic division is a shortcut method used to divide a polynomial by a

1. **State the problem:** Show that there is no real number $x$ such that $$\sqrt{3+x} + \sqrt{7-x} = 5.$$\n\n2. **Domain consideration:** For the expression to be real, both radic

1. **State the problem:** Show that there is no real number $x$ such that $$\sqrt{3+x} + \sqrt{7-x} = 5.$$

1. State the problem: Given the sequence defined by $a_1 = 1$ and the recurrence relation $a_{n+1} = a_n + n + 1$, find a closed-form expression for $a_n$ and compute $a_{100}$. 2.

1. We are asked to simplify the expression $$196n^2 - 144$$. 2. Observe that both terms are perfect squares multiplied by constants: $$196n^2 = (14n)^2$$ and $$144 = 12^2$$.

1. The problem is to evaluate the expression $$3 \ \frac{3}{8} \ 0/0 \left( \frac{5}{12} + \frac{1}{5} \right).$$\n\n2. Notice the term $0/0$ in the expression. Division by zero is

1. **State the problem:** Find the exact value of $\frac{3}{8} \div \left( \frac{5}{12} + \frac{1}{3} \right)$. 2. **Simplify inside the parentheses:** Convert $\frac{1}{3}$ to hav

1. Stated problem: Simplify the algebraic expression $$\frac{196n^2 - 144}{2}$$. 2. Factor the numerator:

1. The problem states we have the curve defined by the function $$y = -x^2 + 5$$ and we want to find where the point at $$x = 3$$ lies on this curve. 2. To find the corresponding $

1. **State the problem:** Factor the quadratic expression $a^2 - 10ab + 21b^2$. 2. **Identify coefficients:** The expression is in terms of $a$ and $b$, and we want to factor it as

1. The problem is to simplify the expression $1 \frac{4}{7} + \frac{2}{3} - 1 \frac{5}{6}$ and then calculate $\frac{\sqrt{27}}{9}$. 2. Convert the mixed numbers to improper fracti

1. **State the problem:** We are given the function $$f(x) = \sqrt{x^2 -4} \times 5 - \sqrt{36 - x^2}$$ and need to find the domain to the right of the origin expressed as $$[2, a)

1. We are given the function $Ri(E) = \frac{N(n,a)(E, ti)}{\Phi(E)}$. 2. To understand this expression, note that it represents a ratio of two quantities: the numerator $N(n,a)(E,

1. **State the problem:** Find the Highest Common Factor (HCF) of $196n^2$ and $144$. 2. **Prime factorize the numerical coefficients:**

1. The problem is to find the Highest Common Factor (HCF) of 196 and 144. 2. List the prime factors of each number.

1. Stating the problem: Solve the equation $4 + 5\sqrt{x} = 2\sqrt{x} + 16$ for $x$. 2. Rearrange the equation to isolate the square root terms on one side and constants on the oth

1. The problem is to simplify the expression $2x^2 - 18$. 2. First, identify the common factor in the terms. Both terms $2x^2$ and $18$ have a common factor of $2$.