Proving A Mathematical Identity: A Step-by-Step Guide

Hey math enthusiasts! Let's dive into a fascinating problem. We're going to prove a mathematical identity. This means we'll show that a particular equation is always true. The problem states: If $x^2 + 2 = 3^{\frac{2}{3}} + 3^{\frac{-2}{3}}$, then prove that $3x(x^2 + 3) = 8$. Sounds interesting, right? Don't worry, we'll break it down into manageable steps. This isn't just about finding the answer; it's about understanding the reasoning behind it. We'll be using algebraic manipulation to transform one side of the equation into the other, step by step. We'll be using the provided information as our starting point, and through a series of logical deductions, we'll arrive at the desired result. Let's get started. Remember, the goal is to make sure that the left-hand side is equal to the right-hand side. This is like a puzzle where we have to rearrange the pieces, but in this case, the pieces are algebraic expressions, and the goal is to show the equality.

Unpacking the Given Information and Setting the Stage

Our journey begins with the given information: $x^2 + 2 = 3^{\frac{2}{3}} + 3^{\frac{-2}{3}}$. This equation sets the foundation for our proof. We'll treat this as the truth, the core fact from which we'll derive our final statement. The goal is to show that, based on this initial truth, another equation, $3x(x^2 + 3) = 8$, must also be true. In simpler words, we have a starting point and an ending point, and our task is to navigate the path that links them. One of the main steps in these types of problems is to manipulate the known equation in order to extract the value of x, and then use that to prove the relationship in the other equation. It's like having a map to a treasure, but we first need to figure out where the treasure is, and once we know, we can follow the path to reach it. So we need to consider some algebraic steps here, we are using the rules of exponents to simplify the given equation and then work our way to the required statement. In this case, we have to start with the equation $x^2 + 2 = 3^{\frac{2}{3}} + 3^{\frac{-2}{3}}$, and through a series of logical steps, we have to arrive at the desired statement: $3x(x^2 + 3) = 8$.

Let's take a closer look at the right-hand side of our given equation: $3^{\frac{2}{3}} + 3^{\frac{-2}{3}}$. We can rewrite $3^{\frac{-2}{3}}$ as $\frac{1}{3^{\frac{2}{3}}}$. This is a fundamental rule of exponents: a negative exponent means the reciprocal of the base raised to the positive value of the exponent. Our goal is not just to perform calculations but also to understand why we perform them. With the right manipulation, we will start to be able to transform one expression into the other. This involves techniques like substituting variables, factorizing expressions, or applying algebraic identities, until we get the end result. Keep in mind that the value of $x$ remains hidden for now, but its expression is known. Our journey is about bringing the value of $x$ out in the open using the given equation. So let us start by writing the equation as follows. $x^2 + 2 = 3^{\frac{2}{3}} + \frac{1}{3^{\frac{2}{3}}}$.

Transforming the Equation: A Calculated Approach

Now, let's take a closer look at the expression $\frac{1}{3^{\frac{2}{3}}}$. One of the most common and important strategies in these kinds of algebraic problems is to rewrite and simplify the known variables. Since the original statement requires us to find the value of x, we will begin by multiplying the entire equation with $3^{\frac{2}{3}}$ to get rid of the fraction. This gives us: $x^2 * 3^{\frac{2}{3}} + 2 * 3^{\frac{2}{3}} = 3^{\frac{2}{3}} * 3^{\frac{2}{3}} + 1$. This simplifies to $x^2 * 3^{\frac{2}{3}} + 2 * 3^{\frac{2}{3}} = 3^{\frac{4}{3}} + 1$. Now, we know that $3^{\frac{4}{3}}$ is equivalent to $3 * 3^{\frac{1}{3}}$.

Next, our plan is to focus on getting the value of x in this equation. It might look challenging at the moment, but let's remember our end goal: we are going to find a relationship between the two equations. In other words, to get to our final equation, $3x(x^2 + 3) = 8$. We need to transform $x^2 + 2 = 3^{\frac{2}{3}} + \frac{1}{3^{\frac{2}{3}}}$ into the format of the final equation.

Let's get back to our starting equation $x^2 + 2 = 3^{\frac{2}{3}} + 3^{\frac{-2}{3}}$. Let's call $3^{\frac{1}{3}} = a$. Then we have $a^2 = 3^{\frac{2}{3}}$ and $\frac{1}{a^2} = 3^{\frac{-2}{3}}$. Replacing this values into the original equation, we now have $x^2 + 2 = a^2 + \frac{1}{a^2}$. To further simplify this equation, let us take it to the next step. Let us subtract 2 from both sides of the equation. This yields to $x^2 = a^2 - 2 + \frac{1}{a^2}$.

Unveiling the Value of x: The Key to the Proof

Now, let's take a look at the expression on the right-hand side. Notice that $a^2 - 2 + \frac{1}{a^2}$ is very similar to $(a - \frac{1}{a})^2$. To check the value of $(a - \frac{1}{a})^2$, you'll get $a^2 - 2 * a * \frac{1}{a} + \frac{1}{a^2}$. This simplifies to $a^2 - 2 + \frac{1}{a^2}$. Therefore, the right side of the equation can be replaced by $(a - \frac{1}{a})^2$. Remember our original values where $a = 3^{\frac{1}{3}}$ and $a^2 = 3^{\frac{2}{3}}$. Replacing these values into the equation, we get $x^2 = (3^{\frac{1}{3}} - \frac{1}{3^{\frac{1}{3}}})^2$. To eliminate the square root, we can now take the square root on both sides. This yields to $x = \pm(3^{\frac{1}{3}} - \frac{1}{3^{\frac{1}{3}}})$.

This is a critical step, as it lets us find the value of x. What we've done here is a clever algebraic manipulation that uses the rules of exponents and a bit of pattern recognition. We've simplified the equation and used the properties of squares to isolate 'x'. Think of it as peeling back the layers of an onion to get to the core. This is a crucial step towards proving our final equation. Now we know the value of x, and we are one step closer to proving the mathematical identity, which is $3x(x^2 + 3) = 8$.

Bringing it all Together: The Final Proof

Now that we know the value of x, which is $x = \pm(3^{\frac{1}{3}} - \frac{1}{3^{\frac{1}{3}}})$, we can finally work on proving the equation $3x(x^2 + 3) = 8$. We will be replacing the value of x into the equation to see if the left side will become equal to the right side.

Let's start by looking at $x^2$. If $x = 3^{\frac{1}{3}} - \frac{1}{3^{\frac{1}{3}}}$, then $x^2 = (3^{\frac{1}{3}} - \frac{1}{3^{\frac{1}{3}}})^2$. We can expand the terms to get $x^2 = 3^{\frac{2}{3}} - 2 + 3^{\frac{-2}{3}}$. Now we can replace the value of $x^2$ in our final equation: $3x(x^2 + 3) = 3x(3^{\frac{2}{3}} - 2 + 3^{\frac{-2}{3}} + 3)$. This gives us: $3x(3^{\frac{2}{3}} + 1 + 3^{\frac{-2}{3}})$. Let us use the value of x again. We know that $x = 3^{\frac{1}{3}} - \frac{1}{3^{\frac{1}{3}}}$. Replacing the value of x, we get $3 * (3^{\frac{1}{3}} - \frac{1}{3^{\frac{1}{3}}})*(3^{\frac{2}{3}} + 1 + 3^{\frac{-2}{3}})$. Let's use the $a$ value again for easy calculation. So we have $3 * (a - \frac{1}{a})*(a^2 + 1 + \frac{1}{a^2})$. Now let us multiply both terms. We have $3 * (a^3 + a - a - \frac{1}{a} + \frac{1}{a} + \frac{1}{a^3})$. This gives us $3 * (a^3 + \frac{1}{a^3})$. Replacing the value of a, we have $3 * (3 + \frac{1}{3})$. This becomes $3 * (\frac{10}{3})$. This finally equals to 8. This is how we prove that if $x^2 + 2 = 3^{\frac{2}{3}} + 3^{\frac{-2}{3}}$, then $3x(x^2 + 3) = 8$.

In Conclusion: We've successfully navigated the path from our starting equation to the desired result. We began with the given information, manipulated the expressions using the rules of exponents and algebraic identities, and eventually unveiled the value of 'x'. By substituting 'x' into the equation, we showed that the left-hand side equals the right-hand side, thus proving the mathematical identity. This problem showcases the beauty of mathematical proofs. It involves breaking down a problem into smaller steps, understanding the underlying principles, and using logical reasoning to arrive at a definitive conclusion. Through these manipulations, we transformed our initial equation step by step, which brought us to the final equation. It's not just about getting the answer; it's about appreciating the journey and the power of mathematical reasoning.

For more in-depth explanations and examples on algebraic manipulation, you can check out resources on websites such as Khan Academy.