🧮 algebra

1. **State the problem:** Calculate the square root of $$18 - 12\sqrt{2}$$. 2. **Rewrite the expression:** We want to find $$\sqrt{18 - 12\sqrt{2}}$$.

1. **Problem 1: Multiple Choice with Justifications** 1) Calculate $|\sqrt{5} - 2| + |2 - \sqrt{5}|$.

1. **State the problem:** Solve the equation $$|4^{y+1} - 1| = |4^y - 3|$$ correct to 3 significant figures. 2. **Rewrite the equation:** Note that $$4^{y+1} = 4 \cdot 4^y$$, so th

1. The problem is to simplify the expression $\sqrt{-x^2}$.\n\n2. Recall that $x^2$ is always non-negative for any real number $x$, so $-x^2$ is non-positive (less than or equal to

1. Problem 12: A car uses 5 liters of gasoline to travel 40 km. How much gasoline is needed to travel 128 km at the same rate? Step 1: Find the rate of gasoline consumption per km.

1. Justify that $$(\sqrt{7} - \sqrt{8})^{11} (\sqrt{7} + \sqrt{8})^{12} = 4\sqrt{14} - 15$$ Step 1: Recognize the conjugate terms and rewrite the expression.

1. **Justify that** $$(\sqrt{7} - \sqrt{8})^{11} (\sqrt{7} + \sqrt{8}) = 4 \sqrt{14} - 15$$ - Start by noting the conjugate multiplication: $$(\sqrt{7} - \sqrt{8})(\sqrt{7} + \sqrt

1. Given the system of equations: $$x + y - z = 2$$

1. **State the problem:** We are given two numbers, $x$ and $y$. The sum of twice the first number and the second number is 22, and subtracting the second number from thrice the fi

1. **Problem b)** Given the inequality $|c - 10| + 12 > 18$, find the interval of $c$. 2. Start by isolating the absolute value term:

1. The problem asks to analyze the algebraic expression $3x + 4$ and answer four questions about it. 2. First, count the number of terms in $3x + 4$. A term is a part of the expres

1. The problem is to factor the given algebraic expressions and identify equivalent expressions. 2. Factor each expression:

1. **State the problem:** We have two functions:

1. **State the problem:** We need to find the current ages of Muneer and Mujeeb given two conditions about their ages in the past and future. 2. **Define variables:** Let $M$ be Mu

1. **Factoring algebraic expressions:** Given: $2x + 10$

1. Let's define variables for the prices: - Let $C$ be the cost of one coat.

1. **Problem:** Find the value of $x$ such that \[ [x \quad 1] \begin{bmatrix} 1 & 0 \\ -2 & 0 \end{bmatrix} = [0 \quad 0] \] 2. **Step 1:** Perform the matrix multiplication:

1. **Stating the problem:** Find the graphs of the functions:

1. Problem (35): Solve the quadratic equation $$-x^2 - 8x + 3 = 0$$. 2. Rewrite the equation as $$-x^2 - 8x + 3 = 0$$ or equivalently $$x^2 + 8x - 3 = 0$$ by multiplying both sides

1. Problem (25): Simplify $\frac{\sqrt{6} \sqrt{24}}{2\sqrt{6}}$ and verify if it equals 12. Step 1: Simplify numerator $\sqrt{6} \sqrt{24} = \sqrt{6 \times 24} = \sqrt{144} = 12$.

1. **Solve the equation** $x^2 - 18 = 0$. 2. Add 18 to both sides: $$x^2 = 18$$