🧮 algebra

1. **Problem statement:** We have a 12-inch by 16-inch piece of cardboard. We cut out squares of side length $x$ from each corner and fold up the sides to form an open box. We want

1. Simplify $\sqrt{50x^3y^2}$. We use the property $\sqrt{a b} = \sqrt{a} \times \sqrt{b}$ and $\sqrt{x^2} = x$ for $x \geq 0$.

1. **State the problem:** Calculate the value of $\frac{262}{10} \times \frac{3600}{4707}$.\n\n2. **Write the expression:** The problem is to multiply two fractions: $$\frac{262}{1

1. The problem is to sketch the graphs of functions, but no specific functions were provided. 2. To sketch a graph, you typically identify key features such as intercepts, extrema

1. Staðfesta jöfnuna: $$4(3x + 4) - 3(x + 7) = 2(x + 16)$$ 2. Dreifðu tölunum yfir svigana með dreifireglu: $$4 \times 3x + 4 \times 4 - 3 \times x - 3 \times 7 = 2 \times x + 2 \t

1. Við erum beðin um að finna gildi stæðunnar $1.0 - 4x$ þegar gefið er að $2x = 6 - x$. 2. Byrjum á að leysa jöfnuna $2x = 6 - x$ fyrir $x$.

1. Við erum beðin um að þáttaðu töluna 420 í frumþætti, þ.e.a.s. að skrifa hana sem margfeldi frumtala. 2. Til að gera þetta byrjum við með að finna frumþætti 420 með því að deila

1. **State the problem:** We have four equations and four graphs labeled i, ii, iii, and iv. We need to match each equation to its corresponding graph. 2. **List the equations:**

1. Problem (03-3-38): Given an infinite decreasing geometric series with sum $S=2.25$ and second term $a_2=0.5$, find the denominator $r$. Formula: For infinite geometric series, $

1. **Problem statement:** We are given a series of logarithmic and exponential expressions with their evaluated results. We will explain how to simplify and evaluate each expressio

1. Problem statement. Problem: $f(x) = x - \sqrt{x^2 - z}$.

1. **State the problem:** We need to divide the mixed number $10 \frac{1}{3}$ by the mixed number $2 \frac{1}{15}$ and express the answer in simplest form. 2. **Convert mixed numbe

1. **Problem Statement:** Find the L.C.M. of the polynomial expressions given in questions 4 and 5. 2. **Formula and Rules:**

1. **State the problem:** We need to find the value of $$(2+i\sqrt{3})^{10} + (2 - i\sqrt{3})^{10}$$. 2. **Identify the formula and approach:** Notice that the two terms are comple

1. **State the problem:** Simplify the expression \n $$\frac{20}{(x-6)(x+6)} - \frac{2}{x-6} \times \frac{1}{4-x}$$\n

1. **State the problem:** We are given the function $f(x) = x - \sqrt{x^2 - 2}$ and want to analyze it. 2. **Understand the domain:** The expression under the square root must be n

1. **Problem statement:** We have 72 technicians and 96 network engineers. We want to arrange them in equal rows such that each row contains only technicians or only network engine

1. **State the problem:** Find the greatest common divisor (GCD) of 96 and 72. 2. **Formula and method:** The GCD of two numbers is the largest positive integer that divides both n

1. **Stating the problem:** The family traveled 400 miles on the first day. We need to find the total length of their road trip and how many miles they still need to travel.

1. **Problem Statement:** We have 72 technicians and 96 network engineers. We want to arrange them in equal rows such that each row contains only technicians or only network engine

1. **State the problem:** We need to analyze the function $$y = \ln^2(x) - 4\ln(x)$$ where $$\ln(x)$$ is the natural logarithm of $$x$$. 2. **Rewrite the function:** Let $$t = \ln(