🧮 algebra

1. **State the problem:** Three housewives A, B, and C jointly purchased a basket of oranges costing 80. We need to find how much each paid.

1. **State the problem:** Three housewives A, B, and C jointly purchased a basket of oranges costing 80. We need to find how much each paid.

1. The problem is to understand polynomials in vector spaces. 2. A polynomial in a vector space is an expression of the form $$p(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n$$ whe

1. **Problem Statement:** We are given a table describing iterations of a fractal-like process with columns: Iteration number $n$, Number of Segments $4^n$, Length of Each Segment

1. The problem is to simplify the expression $M = -2 \times (3,2 - 5,7) + 4 \times (1,6 - 3,1)$. 2. First, perform the subtraction inside each vector:

1. **State the problem:** Divide 0.72 by 0.000144. 2. **Formula used:** Division of decimals can be done by converting to fractions or by moving the decimal point.

1. **Problem 48:** Solve the equation $$\left(\frac{1}{7}\right)^{-2x+3} + 49^{x-1} + 7^{2x-1} = 399$$. 2. **Step 1:** Express all terms with base 7.

1. The problem involves solving multiple exponential and logarithmic equations, and finding domains. 2. For each equation, we use properties of exponents and logarithms:

1. Problem 28: Solve the equation $$9^{x^{2}+1} + 3^{2x^{2} - 1} = \frac{28}{81}$$. 2. Use the fact that $$9 = 3^{2}$$ to rewrite the first term:

1. **Problem 15:** Solve the equation $((0.1(6))^{3x} - 5 = 1296)$. 2. **Problem 16:** Find how much the root of the equation $3^{x+1} \cdot 27^{x-1} = 9^7$ is less than 10.

1. **State the problem:** Solve the inequality $$2\left(\frac{1+\ln(x)}{x}\right) > 0$$ for $x$. 2. **Rewrite the inequality:** Since 2 is positive, the inequality depends on $$\fr

1. **State the problem:** Solve the inequality $$\frac{(x-2)(x+1)}{(x-1)} \leq 0$$ and find the sum of all positive integers satisfying it. 2. **Identify critical points:** The num

1. **State the problem:** We are given the family of second order surfaces (SOS) defined by the equation $$3x^2 + y^2 + 2z^2 + 6xy + 2xz + 4yz + ky + z + 1 = 0$$ and we want to dis

1. **State the problem:** Solve the equation $$\frac{(x-3)(x+1)}{\sqrt{\ln(x-1)}} = 0$$ and find the arithmetic average of its roots. 2. **Understand the equation:** A fraction equ

1. **Problem Statement:** Find the domain and range of the function $$f(x) = \sqrt{7x^2 + 25} + 9$$. 2. **Understanding the function:** The function involves a square root, so the

1. **State the problem:** Find the domain and range of the function $$f(x) = \sqrt{7x^2 + 25} + 9$$. 2. **Domain:** The domain of a function involving a square root requires the ex

1. **Stating the problem:** We are given complex numbers $z_1 = -53i$ and $z_2 = 4 - 3i$. We need to find:

1. **State the problem:** Solve the inequality $$\ln(x-1) + \ln(x-3) > \ln(3)$$ for $x$. 2. **Use logarithm properties:** Recall that $$\ln(a) + \ln(b) = \ln(ab)$$ for $a,b>0$.

1. **State the problem:** Calculate the value of the expression $2\sqrt{0.25} + \sqrt{3}$. 2. **Recall the square root properties:**

1. **State the problem:** Simplify the expression $2\sqrt{x}0.25 + \sqrt{x}3$. 2. **Rewrite the expression:** The expression can be written as $2 \times \sqrt{x} \times 0.25 + \sqr

1. The problem is to solve for $x$ in the equation $ax + b = 0$ where $a$ and $b$ are constants. 2. The formula to solve a linear equation of the form $ax + b = 0$ is: