Solving Exponential Equations: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a fun problem: $\left(\frac{9}{25}\right)^{-\frac{3}{2}}-\frac{\sqrt{x}}{27}=\left(\frac{3^3}{11^2}\right)^{-1}$. Don't worry, it looks a bit intimidating at first glance, but trust me, we'll break it down step by step to find the value of x. This equation combines exponents, roots, and fractions, giving us a great opportunity to flex our algebra muscles. Let's get started, guys!

Understanding the Equation and the Goal

Our main goal here is to solve the exponential equation and, specifically, to isolate x. This means we need to manipulate the equation using the rules of exponents and algebra until x stands alone on one side. This process will involve simplifying exponents, dealing with fractions, and ultimately, finding the value of x that makes the equation true. Before we begin, let's take a closer look at the different parts of the equation to familiarize ourselves with what we are working with. The equation $\left(\frac{9}{25}\right)^{-\frac{3}{2}}-\frac{\sqrt{x}}{27}=\left(\frac{3^3}{11^2}\right)^{-1}$ can be broken down into several key components. First, we have the term $\left(\frac{9}{25}\right)^{-\frac{3}{2}}$. This part of the equation involves a fraction raised to a negative fractional exponent. Next, we have $-\frac{\sqrt{x}}{27}$, which includes the square root of x divided by 27. Finally, we have $\left(\frac{3^3}{11^2}\right)^{-1}$, another fractional term with a negative exponent. Each of these parts needs to be simplified to solve the equation. The negative exponents tell us we'll likely need to flip fractions, and the fractional exponent will turn into a root. We will need to use the rules of exponents, such as $(a/b)^n = a^n / b^n$ and $a^{-n} = 1/a^n$, as well as the properties of radicals. The square root of x can also be represented as $x^{1/2}$. Understanding these different components and how to simplify them is the key to solving the equation. Remember, solving equations is like a puzzle: each step you take brings you closer to finding the final answer, so we will be working step by step to solve the equation.

Now, let's look at a few fundamental concepts that will serve as our guide throughout this journey. The primary concepts involved here are exponents, roots, and basic algebraic manipulation. Remember that a negative exponent means you take the reciprocal (flip) of the base and change the exponent to positive. A fractional exponent like $\frac{3}{2}$ indicates both a power and a root, in this case, a square root. For example, $a^{\frac{3}{2}}$ can be written as $\sqrt{a^3}$. Also, we will use basic algebraic rules like adding, subtracting, multiplying, and dividing both sides of the equation by the same number to isolate x. Always keep the order of operations (PEMDAS/BODMAS) in mind when simplifying expressions. Remember, the goal is always to manipulate the equation to get x by itself on one side. A methodical and step-by-step approach is crucial here. Start by simplifying the exponential and fractional terms, then isolate the term with the square root of x, and finally, solve for x. The key to success is to break down the problem into smaller, more manageable steps, and always double-check your work to avoid any errors.

Simplifying the Exponential Terms

Alright, let's roll up our sleeves and start simplifying those exponential terms. We'll begin with $\left(\frac{9}{25}\right)^{-\frac{3}{2}}$. The negative exponent tells us to flip the fraction. Remember that, so the term becomes $\left(\frac{25}{9}\right)^{\frac{3}{2}}$. Now, the fractional exponent tells us that we'll be dealing with both a power and a root. We can rewrite $\left(\frac{25}{9}\right)^{\frac{3}{2}}$ as $\sqrt{\left(\frac{25}{9}\right)^3}$ or $\left(\sqrt{\frac{25}{9}}\right)^3$. Let's go with the second option. The square root of $\frac{25}{9}$ is $\frac{5}{3}$ because the square root of 25 is 5, and the square root of 9 is 3. So now, we have $\left(\frac{5}{3}\right)^3$. That equals $\frac{5^3}{3^3}$, which is $\frac{125}{27}$. Nice, that one is done! Next, we'll simplify $\left(\frac{3^3}{11^2}\right)^{-1}$. A negative exponent flips the fraction, so we get $\frac{11^2}{3^3}$. We know that $3^3 = 27$ and $11^2 = 121$, thus the term simplifies to $\frac{121}{27}$. We have now simplified both exponential terms. Be sure to carefully and systematically break down each term, using the rules of exponents and roots to transform them into simpler expressions. This process often involves flipping fractions, taking roots, and raising numbers to powers. Keep a close eye on the signs (especially negative exponents) to avoid common mistakes. These simplifications form the foundation upon which the rest of the solution will be built. So, let’s go ahead and substitute these simplified values back into the original equation and move on to the next step, where we will isolate the square root of x.

Now, let's substitute the simplified values back into our original equation. The equation $\left(\frac{9}{25}\right)^{-\frac{3}{2}}-\frac{\sqrt{x}}{27}=\left(\frac{3^3}{11^2}\right)^{-1}$ becomes $\frac{125}{27} - \frac{\sqrt{x}}{27} = \frac{121}{27}$. Our next aim is to isolate the term containing $\sqrt{x}$. To do this, we need to get the term with the square root by itself on one side of the equation. We will then perform algebraic manipulations to achieve this goal, using basic arithmetic operations such as addition and subtraction. Remember that whatever operation we perform on one side of the equation, we must also perform on the other side to maintain the equation's balance. We'll subtract $\frac{125}{27}$ from both sides. That gives us $-\frac{\sqrt{x}}{27} = \frac{121}{27} - \frac{125}{27}$. Simplifying the right side gives us $-\frac{\sqrt{x}}{27} = -\frac{4}{27}$. We are one step closer to isolating x. We are on the right track! In order to remove the fraction in front of the square root, we can multiply both sides of the equation by -27. That gives us $\sqrt{x} = 4$. Remember to keep the signs correct and perform the same operation on both sides of the equation. Always double-check your work to avoid making calculation errors. Now that we have isolated the square root of x, we can proceed to solve for x by getting rid of the square root sign.

Solving for x

We're in the home stretch now, guys! We've simplified the equation to $\sqrt{x} = 4$. To solve for x, we need to get rid of the square root. The inverse operation of taking a square root is squaring. So, we'll square both sides of the equation. This gives us $(\sqrt{x})^2 = 4^2$, which simplifies to $x = 16$. And there you have it! We've solved for x. The value of x that satisfies the original equation is 16. It's always a good idea to check your solution. Plug the value of x back into the original equation to ensure it holds true. If we plug x = 16 into the original equation, we get $\left(\frac{9}{25}\right)^{-\frac{3}{2}}-\frac{\sqrt{16}}{27}=\left(\frac{3^3}{11^2}\right)^{-1}$. We've already determined that $\left(\frac{9}{25}\right)^{-\frac{3}{2}}$ simplifies to $\frac{125}{27}$ and $\left(\frac{3^3}{11^2}\right)^{-1}$ simplifies to $\frac{121}{27}$. Also, $\sqrt{16} = 4$. So, the equation is $\frac{125}{27} - \frac{4}{27} = \frac{121}{27}$, which is true! It shows that we've found the correct value for x. Remember that solving exponential equations like this is all about following the rules of exponents and applying basic algebra to isolate the variable you're trying to solve for. Congratulations on successfully solving this exponential equation! Keep practicing, and you'll become a pro in no time.

Now, let’s revisit the steps in summary. First, we simplified the exponential terms by dealing with negative and fractional exponents. Next, we isolated the square root term. Finally, we solved for x by squaring both sides of the equation. Always remember to double-check your answer by substituting it back into the original equation to ensure that the solution is correct. If you follow these steps and understand the concepts, you can solve similar exponential equations with confidence. Practice makes perfect, and with each problem you solve, your understanding and problem-solving skills will only get better. Always remember the order of operations and the rules of exponents to simplify the problem step by step. Keep going, and keep practicing! If you have any further questions or would like to practice more examples, feel free to ask. Keep up the excellent work, and happy solving!