🧮 algebra

1. **State the problem:** We are given three equations:

1. The problem asks whether the bar (fraction line) in the expression $$\left( \frac{z_1}{z_2} \right) = \frac{a_1 + b_1 i}{a_2 + b_2 i}$$ only affects the numerator. 2. In a fract

1. Let's clarify the concept of conjugates in fractions. 2. The conjugate of a binomial expression like $a + b$ is $a - b$, and vice versa.

1. **State the problem:** Given that $\log 40 = a$, express $\log 4$ in terms of $a$. 2. **Recall the logarithm properties:**

1. The problem states that we have a piece of land in the shape of a rectangle with length $x + 20$ meters and width $x + 5$ meters. 2. We are asked to form a function for the area

1. **Problem:** How many people will arrive in the twentieth group if the first group has 1 person, the second group has 3 people, and each subsequent group has 2 more people than

1. **Problem:** We have groups arriving with the first group having 1 person, the second group 3 people, and each subsequent group having 2 more people than the previous group. Fin

1. **State the problem:** Simplify the expression $$\frac{((-4)^6)^2 \times (-5^2)^3}{(-4)^6 \times (-5)^2}$$. 2. **Recall the exponent rules:**

1. **Problem statement:** Solve the equation step by step. 2. **Step 1: Identify the equation to solve.** Since the user did not specify an equation, please provide the equation yo

1. The problem is to simplify the expression: Quality \left( 1:0.1 \frac{1}{3} \right) \sqrt{gity}. 2. First, interpret the expression inside the parentheses: \(1:0.1 \frac{1}{3}\)

1. **State the problem:** We need to find the product of $2x$ and $3$. 2. **Formula used:** Multiplication of algebraic terms is done by multiplying the coefficients and keeping th

1. **State the problem:** Solve the equation $$4(x + 4) = x + 1$$. 2. **Apply the distributive property:** Multiply 4 by each term inside the parentheses.

1. **Problem:** We have groups arriving with the first group having 1 person, the second group 3 people, and each subsequent group having 2 more people than the previous group. Fin

1. **Problem Statement:** We are running a fund-raising concert. The first group has 1 person, the second group has 3 people, and each subsequent group has 2 more people than the p

1. The problem is to find the determinant of the matrix \(\begin{bmatrix} 8 & 5 \\ -4 & -13 \end{bmatrix}\). 2. The formula for the determinant of a 2x2 matrix \(\begin{bmatrix} a

1. The problem is to find the determinant of the 2x2 matrix $$\begin{pmatrix} 85 & -4 \\ -3 & ? \end{pmatrix}$$. However, the matrix is incomplete as the bottom right element is mi

1. **State the problem:** Expand the cubic polynomial $$f(x) = 2x^3 + 7x^2 + x - 6$$ in powers of $$(x-2)$$. 2. **Recall the binomial expansion formula:** To expand a function arou

1. **Problem:** Find the zeros of the polynomial function $f(x) = 2x^3 - 3x^2 - 8x + 12$ using the Rational Root Theorem. 2. **Rational Root Theorem:** Possible rational roots are

1. **Problem statement:** An article originally priced at $700 is discounted by 20%. We need to find (a) the amount of the discount, (b) the final cost after the discount, and (c)

1. **State the problem:** Calculate the sales tax and total price for a refrigerator priced at 682 with a 7% sales tax.

1. **State the problem:** Solve for $t$ in the equation $$-12 \sqrt{-57t + 64} + 64 + 21 = -75.$$\n\n2. **Simplify the constants on the left side:** $$64 + 21 = 85,$$ so the equati