🧮 algebra

1. **Problem:** Multiply and simplify the radicals for the first expression: $$\sqrt{6}^3 \cdot \sqrt{8}r^2$$ 2. **Recall:** The product of radicals rule: $$\sqrt{a} \cdot \sqrt{b}

1. **State the problem:** Solve the rational equation $$\frac{5}{x+1} + \frac{1}{x-1} = \frac{x}{x-1}$$ for all values of $x$. 2. **Identify restrictions:** The denominators cannot

1. **Problem:** Factor out the GCF from $14r^2s^3 + 20r^3s - 10r^4s^5$. 2. **Step 1:** Identify the GCF of the coefficients: GCF of 14, 20, and 10 is 2.

1. **State the problem:** Solve the rational equation $$1 + \frac{2}{x - 8} = \frac{x}{x^2 - 12x + 32}$$ for all values of $x$. 2. **Factor the denominator on the right side:** Not

1. **State the problem:** We need to find the coordinates of the vertices of the triangle formed by the intersection of the three lines: $$y = x + 1$$

1. **State the problem:** Solve the equation $$5 \left(\frac{x}{3}\right)^2 - 3 \left(\frac{1}{9}\right) = - \frac{8}{9}$$ for $x$. 2. **Recall the formula and rules:** We will sim

1. **State the problem:** We need to find the coordinates of the vertices of the triangle formed by the intersection of the three lines: $$y = x + 1$$

1. **State the problem:** Solve the inequality $3b \leq 9$ and graph it on the number line. 2. **Formula and rules:** To solve inequalities, isolate the variable by performing inve

1. **State the problem:** Solve the equation $ (x - 9)(x + 2) = (x + 4)(x - 7) $ for $x$. 2. **Expand both sides:** Use the distributive property (FOIL) to expand each product.

1. **State the problem:** Solve the inequality $$\frac{x}{4} > 8$$. 2. **Formula and rules:** To solve inequalities involving fractions, multiply both sides by the denominator to e

1. **State the problem:** Solve for $x$ in the equation $4^x = 256$. 2. **Recall the formula and rules:** We know that $a^m = a^n$ implies $m = n$ if the bases $a$ are the same.

1. **State the problem:** Simplify the expression $\log_9 3 + \log_9 27$. 2. **Recall the logarithm property:** The sum of logarithms with the same base is the logarithm of the pro

1. **Problem Statement:** Determine the cost for Nancy to have Saxony carpeting installed in all three bedrooms, given the cost is $7.49 per square foot. 2. **Formula:** Total cost

1. **State the problem:** Simplify the expression $$\frac{8m^6n^2}{4m^3} \cdot \frac{12(mn)^3}{n^2}$$ and write an expression for the area of a rectangle with dimensions $$(2a^2 b)

1. **State the problem:** Simplify the expression $$\frac{\left(-2 m^{-5} n^{4}\right)^{-2}}{\left(3 m^{8} n^{-4}\right)^{-1}}.$$\n\n2. **Recall the rules:**\n- Power of a product:

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

1. **State the problem:** We need to select the equation that best represents the first model (bottom-left) where the scale has one star on the left and three rectangles on the rig

1. **State the problem:** Simplify the expression $$(9m^{\frac{5}{3}})(4m^{-\frac{7}{2}})$$. 2. **Use the product rule for exponents:** When multiplying terms with the same base, a

1. **State the problem:** Simplify the expression $$\frac{\sqrt{y^7} \times \sqrt[5]{y^2}}{\sqrt[5]{y^4}}$$. 2. **Rewrite roots as exponents:** Recall that $$\sqrt{y^7} = y^{\frac{

1. The problem asks to describe each given roster (list of elements) using set-builder notation. 2. Set-builder notation describes a set by stating the properties that its members

1. The problem is to create a table of values and graph the linear function $y = -x + 4$. 2. The formula for a linear function is $y = mx + b$, where $m$ is the slope and $b$ is th