🧮 algebra

1. **State the problem:** Simplify the expression $$\left(6x y^{\frac{1}{2}}\right)^{\frac{2}{4}} \left(8x y^{\frac{3}{2}}\right)^{-4}$$. 2. **Rewrite the exponents:** Note that $$

1. **State the problem:** We need to find an expression for the perimeter of the given cross-shaped figure. 2. **Identify the sides:** The figure has edges labeled with variables:

1. **State the problem:** Find how much longer the length of the rectangular garden is than the width.

1. The problem asks to find $(g - h)(x)$ where $g(x) = 5x^3 + 16x + 7$ and $h(x) = 2x^3 - 5x + 8$. 2. The formula for subtraction of functions is $(g - h)(x) = g(x) - h(x)$.

1. **State the problem:** We need to find the values of $m$ such that the line $y = mx - 6$ is tangent to the curve $y = x^2 - 4x + 3$. Also, find the coordinates of the tangent po

1. **State the problem:** We are given the circle equation $$x^2 + y^2 + 6x - 2y - 26 = 0$$ and the line equation $$y = kx - 5$$. We need to find the values of $$k$$ for which the

1. The problem is to determine if the equation $2x^4 + 6x^2 = 8x^8$ is true for all values of $x$. 2. To check this, we can try to simplify or rearrange the equation and analyze it

1. **State the problem:** Solve the equation $$2x^4 + 6x^2 = 8x^8$$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:

1. The problem asks: 48 is what percent of 94? 2. To find what percent one number is of another, use the formula:

1. **State the problem:** Find what percent 30 is of 69. 2. **Formula:** To find what percent $a$ is of $b$, use the formula:

1. **Stating the problem:** We need to find the domain and range of the function $y = f(x)$ based on the given graph points and arrows. 2. **Understanding domain and range:**

1. The problem asks for the value of the function $f$ at $x=6$, i.e., find $f(6)$. 2. From the description, the function $f$ is graphed as a continuous curve with several points an

1. **Problem Statement:** We need to find a cubic function $f(x)$ that matches the described graph. 2. **Understanding the graph:** The graph is cubic, passing through the origin $

1. **State the problem:** We are given the average time $58.2$ seconds for running $400$ meters, and the times vary by $6.4$ seconds from this average. We need to write an equation

1. **Stating the problem:** Which expression is equivalent to $\frac{3}{5}$? 2. **Understanding the problem:** The fraction $\frac{3}{5}$ means 3 divided by 5.

1. **Problem:** A farmer wants to build a rectangular fence around a garden. The length of the garden is 10 meters more than its width. If the farmer has 60 meters of fencing mater

1. **State the problem:** You have two bundles of spaghetti noodles: one with 5 noodles and one with 10 noodles. The 5-noodle bundle can hold 10 quarters before breaking, and the 1

1. **State the problem:** We need to determine which pair of expressions are equivalent by simplifying each and comparing. 2. **Check pair 1: $4 + 12x$ and $4(3x + 1)$**

1. **State the problem:** Jane starts with 50 in savings and after 8 weeks, she has 290. Each week, she earns some amount $x$ babysitting but donates 5 every week. We need to find

1. **State the problem:** We are given the function $$g(x) = \frac{1}{2} (x + 1)^2 - 3$$ and asked to graph it, identify its domain and range. 2. **Identify the type of function:**

1. **State the problem:** Solve the equation $$3(1 + 11x) = -5(-3 - x) - 12$$ for $x$. 2. **Apply the distributive property:** Multiply out the parentheses on both sides.