🧮 algebra

1. **State the problem:** Simplify the expression $6\sqrt{20} + 7\sqrt{5}$. 2. **Recall the rule:** $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ and simplify square roots by fact

1. **State the problem:** Find the number of digits in the number $5^{22} \times 8^{10}$. 2. **Recall the formula for the number of digits:** The number of digits of a positive int

1. **State the problem:** Simplify the expression $$\left( \frac{49}{p^2} \right)^{\frac{1}{2}}$$. 2. **Recall the rule:** The exponent $\frac{1}{2}$ means taking the square root.

1. **State the problem:** Simplify and solve the expression $$\frac{1}{2} \left[ 8x + 10 - 6 \left( 1 - 4x \right) \right]$$. 2. **Apply the distributive property inside the bracke

1. **State the problem:** Simplify the expression $$\frac{x^{4/5}}{x^{1/5}}$$. 2. **Recall the rule for division of exponents with the same base:** When dividing powers with the sa

1. **State the problem:** Solve the inequality $-2\ln(x-1) \geq 0$. 2. **Rewrite the inequality:** Divide both sides by $-2$. Since dividing by a negative number reverses the inequ

1. **State the problem:** Solve the inequality $-2\ln(x-1) < 0$. 2. **Recall the properties:** The natural logarithm function $\ln(y)$ is defined for $y > 0$.

1. Problem: Given $f(x) = -5x^2 + 10x - 5$, express in factored form, find vertex, zeros, axis of symmetry, direction of opening, domain, and range. Step 1: Factor the quadratic.

1. The problem asks for the value of $f(x)3$, which is ambiguous but likely means $f(x) \cdot 3$ or $f(3)$ depending on context. 2. If $f(x)$ is a function, $f(x)3$ usually means $

1. The problem is to evaluate the expression $\frac{2}{3} \times \frac{82}{7}$.\n\n2. The formula for multiplying fractions is $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b

1. **Problem statement:** Arun's salary is twice Suraj's salary, and Suraj's salary is triple Epsilon's salary. Find Epsilon's annual salary. 2. **Define variables:** Let Epsilon's

1. **Stating the problem:** Solve the system of linear equations: $$5x - 2y = 1$$

1. **Stating the problem:** Two brothers are currently 5 and 8 years old. We want to find in how many years their ages will be in the ratio 4:5. 2. **Formula and explanation:** Let

1. **State the problem:** Simplify the expression $$\frac{\sqrt{a+1} + \sqrt{a-1}}{\sqrt{a+1} - \sqrt{a-1}} - \frac{\sqrt{a+1} - \sqrt{a-1}}{\sqrt{a+1} + \sqrt{a-1}} \div \frac{\sq

1. **Problem statement:** Two cars start from the same point and travel along roads forming a 60° angle. Car 1 travels at 50 km/h and Car 2 at 80 km/h. We need to find:

1. **State the problem:** Solve the equation $32 + 4x = 25 + 5x$ for $x$. 2. **Write down the equation:**

1. Problem: Solve the quadratic equation $x^2 - 5x + 6 = 0$ and find its roots. 2. Formula: The roots of a quadratic equation $ax^2 + bx + c = 0$ are given by the quadratic formula

1. **State the problem:** Solve the equation $x^2 - t = 0$ for $t$ given specific values of $x$. 2. **Formula and explanation:** The equation can be rearranged to find $t$ as $t =

1. **State the problem:** Write the quadratic expression $3x^2 - 24x + 54$ in the form $r(x + p)^2 + q$ where $r$, $p$, and $q$ are integers. 2. **Recall the formula:** The vertex

1. **State the problem:** Write the quadratic expression $x^2 - 13x + 3$ in the form $(x + a)^2 + b$, where $a$ and $b$ are constants. 2. **Recall the formula:** To complete the sq

1. The problem is to solve the equation $x^2 - t = 0$ for $t$ when $x = -4$. 2. The given equation is $x^2 - t = 0$. We want to isolate $t$.