🧮 algebra

1. **State the problem:** Simplify the expression $$\sqrt{\frac{\pi^2}{(\sqrt{2} - 1)(\sqrt{2} + 1)}}$$. 2. **Simplify the denominator:** Notice that $$(\sqrt{2} - 1)(\sqrt{2} + 1)

1. **State the problem:** Simplify the expression $$\frac{1}{\pi} \quad \text{and} \quad \sqrt{\frac{\pi^2}{(\sqrt{2} - 1)(\sqrt{2} + 1)}}.$$\n\n2. **Simplify the denominator insid

1. **State the problem:** We are given the population of a town in 2010 as $1.8 \times 10^{6}$ and in 2012 as $9 \times 10^{3}$. We need to find how many times the population in 20

1. **State the problem:** Find the value of $m$ such that $$(-3)^{m(m+1)} \times (-3)^5 = (-3)^7.$$\n\n2. **Use the property of exponents:** When multiplying powers with the same b

1. **State the problem:** Evaluate the expression $$(5^{-1} \times 2^{-1}) \times 6^{-1}$$. 2. **Recall the meaning of negative exponents:** For any nonzero number $a$, $a^{-1} = \

1. The problem asks to express the size of a plant cell, given as 0.00001275 m, in standard form. 2. Standard form means writing the number as a product of a number between 1 and 1

1. **State the problem:** We are given the function $f(x) = 2x + 3$ and need to find the values of $f(4)$, $f(-6)$, $f(\frac{1}{2})$, and $f(0)$. 2. **Calculate $f(4)$:** Substitut

1. Problem 23: Find the equation of a hyperbola given the latus rectum $l = 10$. The latus rectum of a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is given by $l = \frac{2b^2

1. The equation given is $$-3x^2 + 7y^2 = -1$$. To identify the conic, rewrite it as $$3x^2 - 7y^2 = 1$$ by multiplying both sides by -1.

1. The problem is to analyze the system of equations: $$x + y = 4$$

1. Problem 27: Find the new coordinates of the point (6,7) after rotating the xy-axes through an angle of 45°. Step 1: The rotation formulas for coordinates when axes are rotated b

1. Problem 29: Find the value of $k$ such that the line $y = kx + 1$ is tangent to the hyperbola $3x^2 - 4y^2 = 12$. 2. Substitute $y = kx + 1$ into the hyperbola equation:

1. **State the problem:** Solve the system of equations using substitution and identify the error in the student's solution. Given system:

1. **State the problem:** Given the equation $ (\sin x + \cos x) y = \cos^2 x $ and the value $ x = \frac{\pi}{2} $, find the value of $ y $. 2. **Substitute the value of $ x $:**

1. The problem is to find the value of $w$ in the equation $$\sqrt{82^2 + 15^2 + 144^2} = \sqrt{82^2 + 15^2 + w^2}.$$\n\n2. First, calculate the squares of the known numbers:\n$$82

1. The problem is to simplify the expression $15c10$. 2. Assuming the expression means multiplication of three terms: $15 \times c \times 10$.

1. The problem is to simplify the expression $15c10$. 2. Assuming the expression means multiplication of constants and variables, it can be written as $15 \times c \times 10$.

1. **State the problem:** We analyze the function f represented by curve (C) and line (d) in the given graphs. 2. **Domain of definition of f:** From the graph, the curve (C) is de

1. The problem is to find the formula for the sum of the first $n$ natural numbers, i.e., $s = 1 + 2 + 3 + \cdots + n$. 2. We can use the formula for the sum of an arithmetic serie

1. The problem is to simplify the expression $2C + 7\sqrt{3}C$. 2. Notice that both terms have a common factor $C$.

1. The problem is to divide the mixed number $6\frac{3}{5}$ by the fraction $\frac{9}{10}$. 2. First, convert the mixed number $6\frac{3}{5}$ to an improper fraction.