🧮 algebra

1. **State the problem:** Simplify the expression $3p^6q^{-4} \cdot 5p^{-8}q^7$. 2. **Recall the rules:** When multiplying terms with the same base, add the exponents: $a^m \cdot a

1. **State the problem:** Simplify $$\left( \frac{2p^5}{p^2} \right)^{-5}$$ with positive exponents. 2. **Apply the quotient rule for exponents:** $$\frac{p^5}{p^2} = p^{5-2} = p^3

1. **Problem 9 is not provided in the user's message, so it cannot be answered.** 2. **Problem 10: Find the rate of change in temperature from 6:00 a.m. to 2:00 p.m.**

1. El problema es resolver una ecuación o sistema usando el método de sustitución. 2. El método de sustitución consiste en despejar una variable en una ecuación y luego sustituir e

1. The problem asks to describe the graph of $y = x$ shown with a dotted line and compare it with the solid line graph. 2. The equation $y = x$ represents a straight line passing t

1. The problem states: "Twelve less than -9 times a number is equal to eight minus 4 times the number." We need to translate this into an equation and solve for the number. 2. Let

1. The problem is to understand how to draw a function or graph "over the line," which typically means plotting a function on or above a certain line. 2. To do this, we first ident

1. **Stating the problem:** Solve the system of linear equations: $$\begin{cases} x + y - z = 2 \\ 2x - y + z = 1 \\ x - y + 2z = 2 \end{cases}$$

1. **State the problem:** We need to find the value of $a$ such that the equation $3(a + 5) - a = 23$ is true. 2. **Write the equation:**

1. The problem is to identify which points lie on the graph of the function $y=4x$. 2. The function $y=4x$ means for any $x$ value, $y$ should be $4$ times $x$.

1. **State the problem:** Juan's savings account balance quadrupled during the year. After withdrawing 300, the balance was 400. We need to find the balance before it quadrupled. 2

1. Problem 6: Find the slope of the line passing through points (7, -2) and (3, -2). The slope formula is $$m=\frac{y_2 - y_1}{x_2 - x_1}$$ where $(x_1, y_1) = (7, -2)$ and $(x_2,

1. **State the problem:** We need to find the equation of a line with slope $m=1$ that passes through the point $(-2,5)$. The equation should be in slope-intercept form $y=mx+b$. 2

1. **State the problem:** We are given a function $f(x)$ with points $(1,5)$, $(0,1)$, and $(-1, \frac{1}{5})$. We need to analyze the graph of $g(x)$, describe the transformation

1. **State the problem:** Solve for $x$ in the equation $$2(3x + 8) = 70$$ and simplify the answer. 2. **Apply the distributive property:** Multiply 2 by each term inside the paren

1. **Problem:** Calculate the product $i(3 + 4i)$. 2. **Recall:** Multiplying complex numbers involves distributing and using $i^2 = -1$.

1. The problem asks to simplify and compare the expressions $\frac{1}{8}x + 2$ and $\frac{1}{2}(4 + \frac{1}{4}x)$.\n\n2. Start with the first expression: $\frac{1}{8}x + 2$. This

1. **Stating the problem:** Find the value of the ordered pair $(3,4)$ in the context of algebraic expressions or coordinate points. 2. **Understanding ordered pairs:** An ordered

1. We are asked to simplify the fractions: a) $\frac{15}{35}$, b) $\frac{20}{36}$, c) $\frac{4}{48}$, d) $\frac{75}{90}$, e) $\frac{64}{256}$, f) $\frac{112}{144}$. Simplifying fra

1. **State the problem:** Calculate 55% of 80. 2. **Formula:** To find a percentage of a number, use the formula:

1. **State the problem:** Solve for the unknown in the proportion equations given. 2. **Recall the rule for proportions:** If $\frac{a}{b} = \frac{c}{d}$, then $a \times d = b \tim