🧮 algebra

1. The problem involves a function $f(x)$ and the expression $f(4) - f(-4) = f(b)$. We are asked to find the value(s) of $b$ such that this equality holds. 2. To solve this, we obs

1. **State the problem:** We have a geometric sequence with common ratio $r$. The difference between the 2nd term $a r$ and the 5th term $a r^4$ is 156.

1. **State the problem:** We are given that $x$ and $y$ are positive real numbers and satisfy the system:\n$$\log_5(x+y) + \log_5(x-y) = 3$$\nand\n$$\log_2 y - \log_2 x = 1 - \log_

1. State the problem: Calculate the value of $$M = \left(-\frac{3}{8} \times -\frac{4}{7}\right) \div \left(\frac{3}{5} - \frac{5}{3} + 27 \frac{39}{19}\right)$$. 2. Calculate the

1. Stated problem: Determine if $\frac{-12}{8} = -1.5$ is correct. 2. Perform the division: $\frac{-12}{8} = -1.5$ because dividing -12 by 8 gives -1.5.

1. **Stating the problem:** Solve the equation $$4^{-x+\frac{1}{2}} - 7 \cdot 2^{-x} - 4 = 0$$. 2. **Expressing powers with the same base:** Note that $$4 = 2^2$$, so $$4^{-x+\frac

1. The problem is to evaluate the expression $$-\frac{12}{4} \times \frac{8}{4} = ?$$ 2. First, simplify each division:

1. Start with the original expression: $-12 + \sqrt{144} - 144$. 2. Calculate the square root: $\sqrt{144} = 12$.

1. State the problem: Solve the quadratic equation $3x^2+2x-8=0$ using the factorizing method. 2. Multiply the coefficient of $x^2$ (which is 3) by the constant term (which is -8):

1. The problem asks for the difference in cost between 300 g of potatoes and 300 g of carrots. 2. From the graph, the cost of 300 g of potatoes is approximately $27$ p.

1. The problem is to solve the equation $n^2 = 7^n$ for $n$ using the Lambert W function. 2. Rewrite the equation: $n^2 = 7^n$ implies $n^2 = e^{n \ln 7}$.

1. **State the problem:** We need to find the value(s) of $n$ such that $$n^2 = 7^n$$. 2. **Analyze the equation:** The equation is $$n^2 = 7^n$$. Here, $n^2$ is a quadratic functi

1. Problem: Simplify the expression $\sqrt[3]{4^{1-5}}$. 2. Compute the exponent inside the power: $1-5=-4$.

1. **Problem statement:** Given sets $A = B = \{x \mid -2 \leq x \leq 2\}$ and functions: a) $f(x) = |x|$

1. We are given the quadratic equation $3x^2 + 4x + 1 = 0$, and our goal is to find the values of $x$ that satisfy this equation. 2. Identify the coefficients: $a = 3$, $b = 4$, an

1. Start with the expression: $$2x^{-1/2}$$. 2. Recall that $$x^{-1/2} = \frac{1}{x^{1/2}} = \frac{1}{\sqrt{x}}$$, so the expression is $$2 \times \frac{1}{\sqrt{x}} = \frac{2}{\sq

1. To determine the feasible solutions, we first need the exact question or the system of inequalities/equations given. 2. If the question involves inequalities, the feasible regio

1. Given the equation $$a^{n^{n}} + bn^{n} + c = 0$$, we want to express $$n$$ in terms of the Lambert W function. 2. First, isolate the term containing $$n$$: $$a^{n^{n}} = -bn^{n

1. **Problem statement:** Given two sets $A = B = \{x \mid -2 \leq x \leq 2\}$, determine whether each function is injective, surjective, or bijective.

1. **State the problem:** Solve the quadratic equation $$3n^2 + 4n + 1 = 0$$. 2. **Identify coefficients:** Here, $a = 3$, $b = 4$, and $c = 1$.

1. The problem asks to fully simplify the given expressions.\n\n2. For part (a): Simplify $2k \times 3 \times 5n$.\n- Multiply the constants: $2 \times 3 \times 5 = 30$.\n- Multipl