🧮 algebra

1. **State the problem:** Find the points of intersection between the circle given by the equation $$ (x - 5)^2 + (y - 2)^2 = 36 $$ and the vertical line $$ x = 5 $$. 2. **Use the

1. **State the problem:** Find the slope of the line passing through the points $(1, -7)$ and $(1, 13)$. 2. **Formula for slope:** The slope $m$ between two points $(x_1, y_1)$ and

1. **State the problem:** Divide $\frac{1}{2}$ by $\frac{1}{4}$. Write the answer in simplest form. 2. **Recall the division rule for fractions:** To divide by a fraction, multiply

1. **State the problem:** Solve the equation $$ (3x + 2)^{\frac{2}{5}} = 4 $$. 2. **Formula and rules:** To solve equations with fractional exponents, raise both sides to the recip

1. **Problem 1:** Combine like terms in the expression represented by the bars: $y + y + y + 2$. 2. The like terms are the three $y$s. Adding them gives:

1. **Determine if the two equations are equivalent:** **Problem 1:**

1. **State the problem:** Simplify the expression $$\frac{x^2 - 4x + 3}{x^2 - 9}$$. 2. **Recall the formulas and rules:**

1. **Problem:** Simplify the expression $$\frac{16m^2}{24m^7}$$. 2. **Formula and rules:** When dividing powers with the same base, subtract the exponents: $$\frac{a^m}{a^n} = a^{m

1. **State the problem:** Simplify the expression $$\frac{y^2 - 36}{5y^2 - 26y - 24}$$. 2. **Recall the formula and rules:** To simplify a rational expression, factor both numerato

1. **State the problem:** We need to find which of the given numbers has the greatest value among $4.\overline{2}$, $\sqrt{24}$, $\pi + 1$, and $\frac{9}{2}$.\n\n2. **Convert each

1. The problem asks to identify which of the given inequalities involving square roots is incorrect. 2. Recall that the square root function gives the positive number whose square

1. **State the problem:** Find the product and simplify the expression $$(9m^4)(-8m^3)$$. 2. **Recall the multiplication rules:** When multiplying coefficients (numbers), multiply

1. **State the problem:** Simplify the expression $$3(4u^2 - 1)$$. 2. **Formula and rules:** Use the distributive property, which states that $$a(b + c) = ab + ac$$. Here, multiply

1. **State the problem:** Solve the quadratic equation $x^2 + 6x = 25$. 2. **Complete the square:** To solve by completing the square, add the square of half the coefficient of $x$

1. **State the problem:** Solve the quadratic equation $$x^2 - 10x = -9$$ by completing the square. 2. **Formula and rule:** To complete the square for an equation of the form $$x^

1. **State the problem:** Simplify the expression $$\sqrt{\sqrt{\left(\frac{m^{3}}{m^{-11}}\right)^{4}}}$$. 2. **Use exponent rules:** Recall that $$\frac{a^{x}}{a^{y}} = a^{x-y}$$

1. **State the problem:** Simplify the expression $$\left\{\left[\left(\frac{1}{4}\right)^3 \cdot \left(\frac{2}{3}\right)^3\right]^{-1} \cdot \left(\frac{1}{6}\right)^4\right\}^{-

1. **State the problem:** Simplify each algebraic expression by combining like terms. 2. **Recall the rule:** Like terms have the same variable raised to the same power. Combine th

1. **State the problem:** Factor completely the polynomial $$25x^2 - 9$$. 2. **Formula and rules:** Recognize this as a difference of squares: $$a^2 - b^2 = (a - b)(a + b)$$.

1. **State the problem:** We are given two polynomials:

1. **Problem statement:** A sports car worth 20000 decreases in value by 15% each year. 2. **Modeling the depreciation:** The value after $t$ years is given by the formula for expo