🧮 algebra

1. **Problem a:** Simplify $\left(\frac{a}{b}\right)^{\log 0.5} \cdot \left(\frac{a}{b}\right)^{\log 0.2}$. 2. Use the property of exponents: $x^m \cdot x^n = x^{m+n}$.

1. **State the problem:** Simplify the expression $$\frac{7}{10} - \frac{3}{5} \times \frac{1}{2} + \frac{3}{4}$$. 2. **Recall order of operations:** Multiplication comes before ad

1. **State the problem:** We are given a rectangle with an area of 15 cm². The height is 5 cm and the width is given as $q - 2$ cm. We need to find the value of $q$. 2. **Formula f

1. **State the problem:** Solve the equation $f(|x|) = 0$ where $f(x) = 3|x - 2| - 10$. 2. **Write the equation:**

1. **State the problem:** We want to show mathematically that the line $y=5x+10$ intersects only one branch of the graph of $f(x)=3|x-2|-10$. 2. **Recall the definition of $f(x)$:*

1. **State the problem:** We have a quadratic function $y = x^2 + ax + b$ where $a,b \in \mathbb{Z}$. Points $P(-1,-2)$ and $Q(3,2)$ lie on the curve. We need to find two equations

1. **State the problem:** We have chickens and cows on a farm. Chickens have 2 legs each, cows have 4 legs each, and the total number of legs is 82. 2. **Define variables:** Let $x

1. **State the problem:** We are given the equation of a line $y = -8x + 8$ and a table showing possible combinations of chickens ($x$) and cows ($y$) on a farm, where the total nu

1. **State the problem:** We need to solve the equation $$\frac{2x+4}{x-3} = 3$$ for $x$. 2. **Recall the formula and rules:** To solve a rational equation like this, multiply both

1. The problem is to graph the function $y = \frac{1}{x} + 2$ by starting with the graph of the standard function $y = \frac{1}{x}$ and applying transformations. 2. The standard fu

1. **State the problem:** Solve the equation $$\frac{2}{3}x = 10$$ for $x$. 2. **Formula and rules:** To solve for $x$, we need to isolate $x$ by dividing both sides of the equatio

1. The problem is to analyze the function $$y = \frac{1}{x} + 2$$. 2. This function is a rational function with a vertical asymptote where the denominator is zero, i.e., at $$x=0$$

1. **State the problem:** Solve the equation $$\frac{1}{3}x - \frac{2}{3} = \frac{1}{4}x$$. 2. **Identify the least common multiple (LCM) of denominators:** The denominators are 3

1. The problem asks to analyze the function $h(d) = - \frac{1}{9} (d - 4)^2 + 4$ which models the height of a basketball ball as a function of horizontal distance $d$. 2. This is a

1. The problem gives two functions: $$h(d) = - \frac{1}{9} (d - 4)^2 + 4$$ and $$h(d) = - \frac{1}{9} (d + 1)^2 + 4$$. We are asked to analyze these parabolas, which model the heig

1. **State the problem:** Solve the equation $$3 - \left( \frac{2}{3}x + \frac{1}{9} \right) = x - \left( \frac{x}{3} - \frac{2}{3} \right)$$ for $x$. 2. **Rewrite the equation and

1. **State the problem:** We want to prove that $$2x^2 + 2x > -4$$ for all real values of $$x$$, and then explain why the graph of $$y = 2x^2 + 2x + 4$$ has no x-intercepts.

1. **State the problem:** We have a quadratic function $$y = 3a x^2 - q$$ and it passes through the point $(-1, 6)$. We need to find the coordinates of another point on the parabol

1. The problem is to understand what a mixed number is and how to work with it. 2. A mixed number is a number consisting of an integer and a proper fraction combined, for example,

1. **State the problem:** Simplify the expression $$\frac{-15x^2ab}{12b^2x^2}$$. 2. **Write the formula and rules:** When simplifying fractions with variables and coefficients, div

1. Tehtävä 235: Ratkaise yhtälöt ja tarkista sijoittamalla. Yhtälöiden ratkaisemiseen käytämme toisen asteen yhtälön ratkaisukaavaa: