🧮 algebra

1. **Stating the problem:** Solve an exponential equation that simplifies to a linear equation. 2. **General form:** An exponential equation has the form $$a^{x} = b$$ where $a > 0

1. The problem is to find the general term $u_n$ of the sequence defined by $u_n = 2n + 1$. 2. The formula given is already the explicit formula for the $n$-th term of the sequence

1. **State the problem:** Solve the equation $\left(\frac{1}{8}\right)^{2x} = 4 \sqrt{2}$ for $x$. 2. **Rewrite the bases as powers of 2:**

1. **State the problem:** Given that $3 \sqrt{7} = \sqrt{m}$, find the value of $m$. 2. **Recall the property of square roots:** For any positive numbers $a$ and $b$, $a \sqrt{b} =

1. **State the problem:** Simplify the expression $12xy - 6x + 18yx^2 - 8x - 4xy - 23yx^2$. 2. **Identify like terms:**

1. **State the problem:** We need to find values of $x$, $y$, and $z$ such that the matrices $$\begin{bmatrix} x & 6 \\ -6 & 5 \\ -10 & 5 \end{bmatrix} = \begin{bmatrix} -7 & 6 \\

1. **State the problem:** We have a piecewise function defined as:

1. **State the problem:** Calculate the amount and percent increase in Dale Crosby's salary from 2005 to 2006 using the given line graph. 2. **Identify the salaries for 2005 and 20

1. **Problem Statement:** We are given the equation for the height of an arch above water as a function of horizontal distance $x$ from the river's center:

1. **State the problem:** A motor boat travels 24 km upstream and downstream. The speed of the boat in still water is 18 km/hr. The boat takes 1 hour more to go upstream than downs

1. **State the problem:** Simplify the expression $-2 \times 3x$. 2. **Formula and rules:** When multiplying a constant by a variable, multiply the constants and keep the variable

1. **State the problem:** Simplify the expression $3y \times 5y$. 2. **Recall the multiplication rule for variables:** When multiplying terms with the same base, multiply the coeff

1. **State the problem:** Simplify the expression $5a \times a$. 2. **Recall the rule:** When multiplying powers with the same base, add the exponents: $a^m \times a^n = a^{m+n}$.

1. **Problem:** Given that one root of the quadratic equation $x^2 + x + 5 = 0$ is $-5$, determine which of the following statements is NOT true: 1) The other root is $-5$

1. **Stating the problem:** We are given the function $f(x) = \sqrt[3]{x - 8}$ and want to understand its properties. 2. **Formula and rules:** The cube root function $\sqrt[3]{x}$

1. **State the problem:** Oscar thinks of a number $r$. He squares it to get $r^2$, multiplies by 4 to get $4r^2$, divides by 3 to get $\frac{4r^2}{3}$, then adds 7 to get 55. We n

1. **State the problem:** We have a rectangle and a square with the same perimeter. The rectangle has width $x+7$ cm and height 7 cm. The square has side length $x+5$ cm. We need t

1. **State the problem:** Solve for the two possible values of $p$ in the equation $$\frac{78 - 3p^2}{5} = 6.$$\n\n2. **Use the formula and rules:** To solve for $p$, first elimina

1. **State the problem:** Find the least common multiple (LCM) of the numbers 20, 66, 77, 88, and 99. 2. **Formula and rules:** The LCM of several numbers is the smallest positive

1. **State the problem:** We need to find the equation of a line $L$ that passes through the point $(3, -2)$ and is inclined at an angle of $60^\circ$ to the line $\sqrt{3}x + y =

1. **Problem statement:** Given the quadratic equation $x^2 - bx + c = 0$ with roots $\alpha$ and $\beta$, we need to form a new quadratic equation whose roots are $\alpha + \frac{