🧮 algebra

1. **Problem (ক):** Given $a^x = b$, $b^y = c$, and $c^z = a$, show that $xyz = 1$. 2. **Step 1:** Express $b$ and $c$ in terms of $a$:

1. The problem is to understand why the answer $r = 1.05$ makes sense in the equation $350r = 367.50$. 2. Recall that $r$ represents a multiplier applied to 350 to get 367.50.

1. **Problem statement:** Given the exponential expressions \(P = a^x\), \(Q = a^y\), and \(R = a^z\), prove the following: (ক) If \(a^x = b\), \(b^y = c\), and \(c^z = a\), then s

1. The problem is to find the value of $r$ in the equation $350r = 367.50$. 2. We start with the equation:

1. **Problema:** Resolver la ecuación $2 \log x - 3 + \log \left(\frac{x}{10}\right) = 0$. 2. **Fórmulas y reglas importantes:**

1. **State the problem:** Simplify the expression $4(x-3)-3(x-5)$. 2. **Apply the distributive property:** Multiply each term inside the parentheses by the factor outside.

1. The problem states that $f$ is a one-to-one function with $f(-5) = -11$ and $f(2) = -12$. 2. Recall the definition of an inverse function: if $f(a) = b$, then $f^{-1}(b) = a$.

1. **Problem:** Describe the regions that represent $a^2 - b^2$ in a large square of area $a^2$ containing a smaller square of area $b^2$. Show how to rearrange these regions to il

1. **State the problem:** We are given the function $f(x) = 14 - x^2$ and asked to find its inverse $f^{-1}(x)$, which can be written in two parts. 2. **Find the inverse function:*

1. The problem asks to find the point where the graphs of $y=f(x)$ and $y=f(4x)$ intersect. 2. The intersection means the $x$-values satisfy $f(x) = f(4x)$.

1. **State the problem:** Simplify the expression $$\sqrt[3]{\frac{49}{27}} \cdot j^{\frac{1}{3}}$$ where $j$ is a variable. 2. **Recall the cube root and exponent rules:**

1. The problem is to find the cube root of 64, written as $\sqrt[3]{64}$. 2. The cube root of a number $x$ is a value $a$ such that $a^3 = x$.

1. **State the problem:** Find the domain, vertical asymptote, and end behavior of the function $$h(x) = -\log(3x - 4) + 7$$. 2. **Domain:** The logarithm is defined only when its

1. **State the problem:** Two angles are supplementary, meaning their measures add up to 180 degrees. One angle is twelve degrees more than twice the other angle. 2. **Define varia

1. **Problem Statement:** We have a bacteria population starting at 500 and doubling every 3 hours. We want to write an exponential function to model the population after $t$ hours

1. **Problem statement:** Find all real zeros of the function $$f(x) = x^3 - 6x^2 + 11x - 6$$ and determine if any zeros have multiplicity greater than one. 2. **Formula and rules:

1. Let's start by stating the problem: We want to factor a function, which means expressing it as a product of simpler functions or polynomials. 2. The general formula or approach

1. **State the problem:** Simplify the expression $$\sqrt[3]{\frac{49}{27}} \cdot j^{\frac{1}{3}}$$ where $j$ is a variable. 2. **Recall the cube root and exponent rules:**

1. **State the problem:** Simplify the expression $$\sqrt[3]{\frac{49}{27}} j^{\frac{1}{3}}$$. 2. **Recall the cube root and exponent rules:**

1. Das Problem lautet: Löse die Gleichung oder Aufgabe, die du gestellt hast. Da keine spezifische Aufgabe gegeben wurde, erkläre ich, wie man eine allgemeine lineare Gleichung lös

1. The problem is to determine if the inequality $67.95 < u < 73.25$ is correct. 2. This inequality means that the variable $u$ is greater than 67.95 and less than 73.25.