🧮 algebra

1. **State the problem:** Simplify the expression $$\frac{\sqrt{4x^{6}y^{-2}} \times (x^{2})^{-2}}{(2x)^{0} \times y^{-3}}$$

1. The problem asks to find the value that completes the square for each quadratic expression and rewrite it as a perfect square. 2. The formula to complete the square for an expre

1. **Problem statement:** Graph the function $y=3x^2$ and find the equation of the tangent line to the curve at the point where $x=1$. 2. **Formula and rules:** The slope of the ta

1. **Problem 2:** Solve the equation $ (m - 7)(m + 1) = 0 $ by factoring. 2. **Problem 4:** Solve the equation $ (6 - 6)(b + 3) = 0 $ by factoring.

1. The problem asks to find the rate of change of Arturo's savings over time. 2. The rate of change between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

1. **Stating the problem:** We have the linear function $y = 36.9x$ which represents the relationship between gallons of gas ($x$) and miles driven ($y$). 2. **Formula and explanat

1. Solve $n^2 + 10n + 26 = 5$ for $n$. Rewrite as $n^2 + 10n + 21 = 0$.

1. **State the problem:** We are given a table showing the number of miles $y$ Parker can drive for every $x$ gallons of gas. We need to find the rate of change, which represents h

1. **State the problem:** We are given a table showing Avery's earnings $y$ for working $x$ hours. We need to find the rate of change, which represents how much Avery earns per hou

1. **State the problem:** Given the equation $$a \cos^{12} A + b \cos^{8} A + c \cos^{6} A - 1 = 0$$ and the condition $$\sin A + \sin^2 A = 1,$$ find the value of $$\frac{b+c}{a+b

1. The problem states the ratio of red cars to black cars is 79 : 51. 2. We want to express this ratio in the form $n : 1$, where $n$ is a decimal number.

1. **State the problem:** Solve for $m$ in the equation $$3(2m + 5) = 4m + 29$$. 2. **Use the distributive property:** Multiply 3 by each term inside the parentheses:

1. Problem: Add $\frac{5}{8} + \frac{1}{4}$ and simplify. Formula: To add fractions, use $$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$ where $a,b,c,d$ are integers and $b,d \n

1. The problem is to understand what a logarithm is and how to work with it. 2. A logarithm answers the question: to what power must we raise a base number to get another number? T

1. **State the problem:** We have a rectangle and a square with the same perimeter. The rectangle has length $x+10$ cm and height 4 cm. The square has side length $x+4$ cm. We need

1. The problem is to simplify the equation given by the user. 2. Since no specific equation was provided, simplification generally involves combining like terms, factoring, expandi

1. Problem 10: Calculate the highest temperature recorded at Scott Base. The problem states the highest temperature recorded is 63.8 °C.

1. **State the problem:** We want to simplify the expression $$(A+B+C)(A+B+C)(A+B+C)$$ using algebra. 2. **Formula and rules:** This expression is the cube of a trinomial, written

1. **State the problem:** We are given the function $f(x) = x^3 - 3x^2 - 9x - 5$ and need to find the $x$-intercepts and $y$-intercept. 2. **Recall definitions:**

1. **State the problem:** Simplify each rational expression and state its domain. 2. **Recall:** The domain excludes values making the denominator zero.

1. **State the problem:** Simplify the expression $$(ab^2 c)^3 (2a^2)^2$$ and compare it with the expression $$2a^5 (b^3 c^2)^2$$ to identify which option (A, B, C, or D) matches.