🧮 algebra

1. **State the problem:** We need to find the smallest positive integer $n$ such that there exists a positive integer $k$ where the inequality $$\frac{7}{13} < \frac{n}{n+k} < \fra

1. **State the problem:** Simplify the expression $$\sqrt{3 + 2\sqrt{2}} + \sqrt{3 - 2\sqrt{2}}$$. 2. **Analyze each term:** The expressions inside the square roots involve nested

1. We are given the function $$f(n) = \frac{(\frac{100}{3})^n}{n!}$$

1. **Problem statement:** Find positive integers $a$ and $b$ with $a \leq b$ satisfying $$a^2 + b^2 + 3ab = 719$$

1. **State the problem:** Find the value of $$\frac{300^2}{253^2 - 247^2}$$. 2. **Recognize the denominator as a difference of squares:**

1. Define the concepts: (i) Domain is the set of all possible input values (x-values) for which the function is defined.

1. The problem is to simplify each given fraction expression step-by-step. 2. For expression 1: $\frac{a}{b} + \frac{c}{d}$, find common denominator $bd$, then write $$\frac{ad}{bd

1. Sederhanakan bentuk berikut dan selesaikan: (a) $4^5 \times 4^3 \times 4^{-6} = 4^{5+3-6} = 4^{2} = 16$

1. **State the problem:** Calculate the sum of $6 \times 10^{1}$ and $9 \times 10^{1}$.\n\n2. **Evaluate each term:**\n\n- $6 \times 10^{1} = 6 \times 10 = 60$.\n\n- $9 \times 10^{

1. **State the problem:** We need to add two numbers given in scientific notation: $5.2 \times 10^7$ and $3.01 \times 10^4$. 2. **Rewrite the numbers with the same power of 10:** T

1. Stating the problem: An investor invested a total amount $T$ in two ventures with the same percentage return per year. From the total amount, $\frac{3}{10}T + 600$ was invested

1. **State the problem:** Solve the rational inequality $\frac{2x - 1}{x - 2} \leq 3$.\n\n2. **Rewrite the inequality:** Move all terms to one side to have zero on the other side:\

1. Problem: After $t$ seconds, the height $h$ of an object thrown upward is given by $$h = -16t^2 + 96t.$$ Find $t$ when $h=144$ feet.

1. **State the problem:** Simplify the expression $$\frac{24r^{20}}{3r^2 \times 10r^6}$$ and also simplify $$\frac{6f^7 \times 4f^2}{8f^{14}}$$. 2. **Simplify the first expression:

1. Problem 28: A total of 35000 is invested at 4%, 5%, and 6% interest rates. The first year's interest is 1780.

1. The problem is to simplify the expression $$10y^6 \div 2y^3$$. 2. Start by rewriting the division as a fraction:

1. **State the problem:** We have a pyramid of bricks where the top brick expression equals the sum of the two bricks below it. The expressions are:

1. State the problem: Simplify the expression $$\frac{3a^3}{12a^9}$$. 2. Simplify the coefficients: $$\frac{3}{12} = \frac{1}{4}$$.

1. Problem h) Solve for $y$ in the equation $6(y - 3) - 5(y - 8) = 48 - 3(y)$. 2. Expand both sides:

1. **State the problem:** We need to find the value of $a$ given the equation $a + 345 = 700$. 2. **Isolate $a$:** Subtract 345 from both sides of the equation to get $a = 700 - 34

1. Problem: Solve for $x$ in the equation $6x^2 = 81$. Step 1: Divide both sides by 6: $x^2 = \frac{81}{6} = 13.5$.