🧮 algebra

1. **Problem:** Simplify the expression $$\frac{n!}{(n-1)!}$$ and identify the correct answer from the options. 2. **Recall the factorial definition:**

1. **Problem 12: Recursive definition of factorial** The factorial function $n!$ is defined recursively. We need to identify the correct recursive definition.

1. Problem 19: Identify the correct statement for the inductive step in mathematical induction. The inductive step shows that if the statement holds for some integer $k$, then it a

1. **State the problem:** Calculate the value of $\frac{10!}{8!}$. 2. **Recall the factorial definition:**

1. **State the problem:** Rewrite the quadratic expression $x^2 - 6x + 1$ in the form $(x + a)^2 + b$ where $a$ and $b$ are integers.

1. **Problem Statement:** Find the cardinality of the set $S$ of all distinct numbers of the form $\frac{p}{q}$ where $p, q \in \{1, 2, 3, 4, 5, 6\}$. 2. **Understanding the proble

1. **State the problem:** Simplify the expression $$(5g+2)(g-1)-(3+g)(2g-7)$$. 2. **Use the distributive property (FOIL) to expand each product:**

1. **Problem:** Find the value of $x$ if $16^{245} = 32^x$. 2. **Formula and rules:** Express both sides with the same base. Note that $16 = 2^4$ and $32 = 2^5$.

1. **Solve the equation** Given:

1. **State the problem:** Simplify the expression $$\frac{4x}{y} + \frac{6}{y}$$. 2. **Identify the common denominator:** Both terms have the denominator $y$.

1. **Problem:** Find $\text{Fib}(10)$ given $\text{Fib}(9) = 34$ and $\text{Fib}(8) = 21$. 2. **Formula:** The Fibonacci sequence is defined as:

1. **Problem:** Find the value of $15 \div (125)^{1/3}$. 2. **Formula and rules:** The cube root of a number $a$ is written as $a^{1/3}$. We simplify the cube root first, then perf

1. **State the problem:** Calculate the value of $-204$ dBW/Hz $+ 8$ dB $+ 10 \log 2000000$. 2. **Recall the formula:** The expression involves logarithms and addition of decibel v

1. The problem is to evaluate the expression $10\log 290$. 2. Recall that $\log$ without a base specified usually means base 10 logarithm.

1. **State the problem:** Solve the expression $141 \times 9621 / 141 + (9621 - 1)$. 2. **Recall the order of operations:** Perform multiplication and division from left to right,

1. **State the problem:** Solve for the slope and intercept of the equation $$3y - 3y = -9$$. 2. **Simplify the equation:** Notice that $$3y - 3y = 0$$, so the equation becomes:

1. **State the problem:** We need to find the slope and y-intercept of the line given by the equation $$2y + 6 = 4x$$. 2. **Rewrite the equation in slope-intercept form:** The slop

1. **State the problem:** Solve for $x$ in the equation $$\sqrt{5x+4} - \sqrt{x} = 4.$$\n\n2. **Isolate one square root:** Add $\sqrt{x}$ to both sides to get $$\sqrt{5x+4} = 4 + \

1. **State the problem:** Find the first 4 terms of the expansion of $ (1-8x)^3 $ in ascending powers of $ x $, then use $ x=\frac{1}{100} $ to evaluate $ \sqrt[3]{23} $ correct to

1. **State the problem:** Prove that $a^{\log_a x} = x$ for $a > 0$, $a \neq 1$, and $x > 0$. 2. **Recall the definition of logarithm:** The logarithm $\log_a x$ is defined as the

1. **State the problem:** Solve the inequality $|x-2| > 5$. 2. **Recall the definition of absolute value inequality:** For $|A| > B$ where $B > 0$, the solution is $A < -B$ or $A >