🧮 algebra

1. **State the problem:** We are given the number of infections for weeks 0 to 3 and told the infections grow at a rate of 50% per week. We want to find the number of infections at

1. **State the problem:** We need to find the sum of the infinite geometric series $$5 + 2.5 + 1.25 + 0.625 + ...$$ 2. **Identify the first term and common ratio:**

1. **State the problem:** We need to find the value of the function $f(x) = 4 \cdot 3^x$ when $x = -1$. 2. **Recall the formula:** The function is given by $$f(x) = 4 \cdot 3^x$$

1. **State the problem:** We are given the function $f(x) = \sqrt{4 - x}$ and need to find its inverse function $f^{-1}(x)$. 2. **Recall the definition of inverse function:** The i

1. **State the problem:** Tracy takes 8 hours to complete the task alone, Dave takes 12 hours alone. We want to find how long they take working together. 2. **Formula used:** When

1. **State the problem:** Solve the equation $$5^{2x} = 25^{4 - x}$$ for $x$. 2. **Rewrite the bases:** Note that $25 = 5^2$, so rewrite the right side:

1. **State the problem:** We want to find the time $x$ (in years) it takes for 15 grams of an element to decay to 3 grams, given a constant decay rate of 3% per year. 2. **Formula

1. **State the problem:** We want to find the number of bacteria after 6 hours given an initial amount and a growth rate. 2. **Formula:** The exponential growth formula is $$y = y_

1. **State the problem:** We want to find the time it takes for Jane and Bill to clean the park together. 2. **Formula used:** When two people work together, their combined work ra

1. **State the problem:** We are given the first 5 terms of an infinite geometric sequence: $100$, $60$, $36$, $21.6$, and $12.96$. We need to find the sum of all terms in this inf

1. **State the problem:** We need to find the sum of the first 15 terms of a geometric sequence where the first term $a = 7$ and the common ratio $r = -3$. 2. **Formula for the sum

1. **State the problem:** We are given the first 5 terms of a geometric sequence: $12, 6, 3, \frac{3}{2}, \frac{3}{4}$. We need to find the formula for the $n^{th}$ term, $a_n$. 2.

1. **State the problem:** We need to solve the logarithmic equation $$\log_5(-3x + 5) = 3$$ for $x$. 2. **Recall the definition of logarithm:** The equation $\log_b(a) = c$ is equi

1. **State the problem:** We want to find the time $t$ in weeks when the number of infected people $y$ reaches 1458, given the equation $$y = 2(3^t).$$ 2. **Write the equation with

1. **State the problem:** We need to find the value of $\log\left( \frac{x^{2}y^{3}}{z} \right)$ given $\log(x) = 3$, $\log(y) = 2$, and $\log(z) = -1$. 2. **Recall the properties

1. **State the problem:** Find the value of $$\log\left( \frac{x^2 y^3}{z} \right)$$ given $$\log(x) = 3$$, $$\log(y) = 2$$, and $$\log(z) = -1$$.

1. **State the problem:** We want to simplify the expression $$\log_5(3) + \log_5(8)$$ using properties of logarithms. 2. **Recall the property:** The sum of logarithms with the sa

1. **State the problem:** We need to solve the logarithmic equation $$\log_5(125) = x$$. 2. **Recall the definition of logarithm:** The logarithm $$\log_b(a) = c$$ means $$b^c = a$

1. **State the problem:** We need to analyze and graph the exponential function $$f(x) = -6\left(\frac{4}{3}\right)^x$$ and answer questions about its behavior.

1. **State the problem:** Solve the exponential equation $$25^{x+1} = 5^{-x+11}$$ for $x$. 2. **Rewrite the bases:** Note that $25$ and $5$ are related since $25 = 5^2$. So rewrite

1. **State the problem:** We need to find the value of the exponential function $f(x) = 4 \cdot 3^x$ when $x = 2$. 2. **Recall the formula:** The function is given by $$f(x) = 4 \c