🧮 algebra

1. **Problem:** Evaluate $4 + 5^2 \times 7$. 2. **Order of operations:** Use PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtractio

1. **State the problem:** Find the infimum of the set $$\left\{ \frac{m+n}{mn} \mid m,n \in \mathbb{N} \right\}$$ where $m$ and $n$ are natural numbers. 2. **Rewrite the expression

1. **State the problem:** We need to find the solutions for each equation and then arrange the equations in increasing order of their solution values. 2. **Equation 1:** $$\frac{1}

1. The problem asks to evaluate the expression $2^2$. 2. Recall that $a^n$ means multiplying $a$ by itself $n$ times.

1. The problem is to understand the set $\mathrm{imf} = \left\{ \frac{m+n}{mn} \mid m,n \in \mathbb{N} \right\}$. 2. This set consists of all numbers that can be expressed as the s

1. **State the problem:** Simplify the expression $\frac{1}{4} \times \frac{5}{4}$.\n\n2. **Formula used:** When multiplying fractions, multiply the numerators together and the den

1. **State the problem:** Solve the equation $$\frac{5}{2}x - 7 = \frac{3}{4}x + 14$$ for $x$. 2. **Write down the formula and rules:** To solve for $x$, we want to isolate $x$ on

1. **State the problem:** Justin weighs 15 pounds less than Greg. Half of Greg's weight is 75 pounds less than Justin's weight. We need to find their weights.

1. **State the problem:** Solve the equation $2^{4y} + 1 - 3 = 0$ for $y$. 2. **Simplify the equation:** Combine like terms.

1. **State the problem:** Solve the equation $2^{4y} + 1 - 3 = 0$ for $y$. 2. **Simplify the equation:** Combine like terms on the left side.

1. The problem is to find the other solution of an equation given that one solution is $(3,3)$ and it is incorrect. 2. Since the user did not specify the equation, let's assume it

1. **State the problem:** Solve the equation $24y + 1 - 3 = 0$ for $y$. 2. **Simplify the equation:** Combine like terms on the left side.

1. The problem is to find the points where the function $f(x) = |x| \left\lfloor \frac{1}{x} \right\rfloor$ is split or changes behavior. 2. The function involves the absolute valu

1. **State the problem:** Solve the equation $24y + 1 - 3 = 0$ for $y$. 2. **Simplify the equation:** Combine like terms on the left side.

1. The problem is to multiply the mixed numbers $-2 \frac{1}{2}$ and $3 \frac{2}{5}$ and express the answer as a simplified mixed number. 2. First, convert the mixed numbers to imp

1. **State the problem:** Simplify the expression $$-\frac{3i}{72}$$ and write the answer in the form $a + bi$. 2. **Simplify the fraction:** The fraction $$-\frac{3i}{72}$$ can be

1. **State the problem:** Simplify the expression $$\frac{1 - i}{4 + 4i}$$ and write the answer in the form $a + bi$ with reduced fractions. 2. **Recall the formula:** To simplify

1. **Problem:** Find the range of the function $f(x) = |x|$. 2. **Formula and rules:** The absolute value function $|x|$ outputs the distance of $x$ from zero on the number line, w

1. The problem asks why the function $f(x) = (x-1)e^x$ is not periodic. 2. A function $f(x)$ is periodic if there exists a positive number $T$ such that $f(x+T) = f(x)$ for all $x$

1. **State the problem:** Solve the equation $r^2 \sin 2\theta = 2a^2$ for $r$ in terms of $\theta$ and $a$. 2. **Recall the formula and rules:** The equation involves polar coordi

1. The problem asks to simplify or express the given algebraic expressions involving $f(x)$ and polynomials in $x$. 2. We are given four expressions: