🧮 algebra

1. Given the expression to simplify: $$ (36c^6 d^4)^{\frac{1}{2}} $$. 2. Recognize that raising to the power of $\frac{1}{2}$ means taking the square root: $$ \sqrt{36c^6 d^4} $$.

1. Stated problem: Solve the cubic equation $2x^3 - 2x^2 - 4 = 0$. 2. Factor out the common factor of 2:

1. On considère la suite $S_n = \sum_{k=1}^n \frac{k}{2^{k-1}} = 1 + \frac{2}{2} + \frac{3}{2^2} + \cdots + \frac{n}{2^{n-1}}$ définie pour $n \in \mathbb{N}^*$. 2. Montrons la for

1. **State the problem:** Simplify the expression $$\frac{4x^{18} y^8 \times 6x^9 y^{16}}{(2xy^2)^3}$$ and express it in the form $$ax^b y^c$$ where $a$, $b$, and $c$ are constants

1. We are given the matrix equation: $$\begin{pmatrix} x & s - x^2 \\ -3 & 1 \end{pmatrix} \begin{pmatrix} x - 2 & 1 \\ 2 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 9 \\ 6 & c \end{pm

1. **State the problem:** We need to find the average amount of money spent per customer in May. 2. **Given data:**

1. State the problem: Simplify and solve the equation $$\frac{4x^2}{40x} = 3$$. 2. Simplify the left-hand side by dividing numerator and denominator:

1. We are given the equation $(2x^3)^a = bx^{12}$ and need to find the values of $a$ and $b$. 2. First, apply the power to both the coefficient and variable inside the parentheses:

1. Stating the problem: Simplify the expressions for $A$, $B$, and then calculate $C = A(\sqrt{6} - B)$, where $A = \sqrt{150} - 2\sqrt{24} + \sqrt{36}$ and $B = 3\sqrt{8} \times \

1. **State the problem:** Simplify expressions for $A$, $B$, and $C$ given: $$A = \sqrt{150} - 2\sqrt{24} + \sqrt{36}$$

1. We start with the problem: a) Calculate $ (1.8 \times 10^9) + (6.2 \times 10^7) $ in standard form.

1. State the problem: Calculate \( \frac{2.45 \times 10^{-5} \times 4.76 \times 10^{12}}{8.06 \times 10^{3}} \) and express the answer in standard form to 3 significant figures. 2.

1. We are given the expression $$\frac{2.45 \times 10^{-5} \times 4.76 \times 10^{12}}{8.06 \times 10^{3}}$$ and asked to simplify it and express the answer in standard form with 3

1. **State the problem:** We need to find the values of $a$ and $b$ in the linear equation $$C = aN + b$$ that models the cost, $C$, in pounds, of making $N$ loaves of bread when $

1. Stated problem: Solve the system of simultaneous equations: $$x + y = 5$$

1. **State the problem:** We need to find the equation of a straight line that is parallel to line A and passes through point P(0, 2).

1. We are given the system of simultaneous equations: $$y-2x=5$$

1. Stating the problem: Find the domain and range of the function $$y=\sqrt{x^2-3x}$$.\n\n2. Domain: The expression under the square root must be non-negative for real values of $y

1. The problem states that -8, x, y, 19 form an arithmetic progression (AP). 2. In an AP, the difference between consecutive terms is constant, so the common difference $d$ satisfi

1. Stating the problem: Solve for $x$ in the equation $4^{-3x+2} = 128$. 2. Rewrite $128$ as a power of $4$. Since $4 = 2^2$, and $128 = 2^7$, rewrite both bases as powers of $2$:

1. Stating the problem: We are given four terms in an arithmetic progression (AP): 8, x, y, 19. 2. Recall that in an AP, the difference between consecutive terms is constant. Let t