Solving Exponential Equations: A Step-by-Step Guide

Let's dive into solving the exponential equation $(\sqrt{\frac{\sqrt[4]{16^{12}}}{16^2}})^x=16^3$. Exponential equations might seem daunting at first, but with a systematic approach and a solid understanding of exponent rules, they become much more manageable. In this comprehensive guide, we will break down each step, ensuring you grasp the underlying concepts and can confidently tackle similar problems. So, if you're ready to master the art of solving exponential equations, let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand the problem. We are given the equation $(\sqrt{\frac{\sqrt[4]{16^{12}}}{16^2}})^x=16^3$ and our goal is to find the value of $x$ that satisfies this equation. To do this, we'll need to simplify the equation using exponent rules and algebraic manipulations.

Exponential equations are equations where the variable appears in the exponent. Solving them often involves expressing both sides of the equation with the same base, then equating the exponents. This strategy relies on the fundamental property that if $a^m = a^n$, then $m = n$, provided $a$ is not 0, 1, or -1.

Key Concepts and Rules

To solve this equation effectively, we'll utilize several key concepts and exponent rules:

1. Exponent Rule: $(a^m)^n = a^{mn}$
2. Exponent Rule: $\frac{a^m}{a^n} = a^{m-n}$
3. Root as Fractional Exponent: $\sqrt[n]{a} = a^{\frac{1}{n}}$
4. Rewriting Numbers as Powers of a Common Base: Expressing numbers as powers of a common base is crucial for simplifying exponential equations. In this case, since we have 16, we can express it as $2^4$.

These rules will be our guiding principles as we navigate through the equation. Let's keep them in mind as we begin to simplify the equation step by step.

Step-by-Step Solution

Now, let's break down the solution into manageable steps. We will start by simplifying the expression inside the parentheses and work our way outwards.

1. Simplify the Innermost Term: $\sqrt[4]{16^{12}}$

Our first task is to simplify the term $\sqrt[4]{16^{12}}$. We can rewrite this using the root as a fractional exponent rule:

$\sqrt[4]{16^{12}} = (16^{12})^{\frac{1}{4}}$

Now, apply the exponent rule $(a^m)^n = a^{mn}$:

$(16^{12})^{\frac{1}{4}} = 16^{12 \cdot \frac{1}{4}} = 16^3$

So, the innermost term simplifies to $16^3$. This is a significant simplification, as it makes the expression much easier to handle.

2. Simplify the Fraction: $\frac{16^3}{16^2}$

Next, we need to simplify the fraction $\frac{16^3}{16^2}$. We can use the exponent rule $\frac{a^m}{a^n} = a^{m-n}$:

$\frac{16^3}{16^2} = 16^{3-2} = 16^1 = 16$

This simplifies the fraction to just 16. Now our equation looks much cleaner.

3. Simplify the Square Root: $\sqrt{16}$

Now we have $\sqrt{16}$, which is a straightforward simplification:

$\sqrt{16} = 4$

So, the expression inside the parentheses simplifies to 4. Our equation is now $(\sqrt{16})^x = 4^x = 16^3$.

4. Rewrite with a Common Base

To solve for $x$, we need to express both sides of the equation with the same base. We know that $4 = 2^2$ and $16 = 2^4$, so we can rewrite the equation as:

$(2^2)^x = (2^4)^3$

Apply the exponent rule $(a^m)^n = a^{mn}$:

$2^{2x} = 2^{4 \cdot 3} = 2^{12}$

Now both sides of the equation have the same base, which is 2. This is a crucial step in solving exponential equations.

5. Equate the Exponents

Since the bases are the same, we can equate the exponents:

$2x = 12$

6. Solve for $x$

Finally, solve for $x$ by dividing both sides by 2:

$x = \frac{12}{2} = 6$

Therefore, the solution to the equation $(\sqrt{\frac{\sqrt[4]{16^{12}}}{16^2}})^x=16^3$ is $x = 6$.

Alternative Approach: Converting to Base 2 Early

Another effective method is to convert all terms to a common base (in this case, 2) right from the start. This approach can sometimes simplify the calculations and make the solution process clearer.

Let's revisit the original equation: $(\sqrt{\frac{\sqrt[4]{16^{12}}}{16^2}})^x=16^3$

1. Convert 16 to $2^4$

Replace every instance of 16 with $2^4$:

$(\sqrt{\frac{\sqrt[4]{(2^4)^{12}}}{(2^4)^2}})^x = (2^4)^3$

2. Simplify Exponents

Apply the exponent rule $(a^m)^n = a^{mn}$:

$(\sqrt{\frac{\sqrt[4]{2^{48}}}{2^8}})^x = 2^{12}$

3. Simplify the Fourth Root

Rewrite the fourth root as a fractional exponent:

$\sqrt[4]{2^{48}} = (2^{48})^{\frac{1}{4}} = 2^{48 \cdot \frac{1}{4}} = 2^{12}$

Now the equation looks like this:

$(\sqrt{\frac{2^{12}}{2^8}})^x = 2^{12}$

4. Simplify the Fraction

Use the exponent rule $\frac{a^m}{a^n} = a^{m-n}$:

$\frac{2^{12}}{2^8} = 2^{12-8} = 2^4$

So the equation becomes:

$(\sqrt{2^4})^x = 2^{12}$

5. Simplify the Square Root

Rewrite the square root as a fractional exponent:

$\sqrt{2^4} = (2^4)^{\frac{1}{2}} = 2^{4 \cdot \frac{1}{2}} = 2^2$

Now the equation is:

$(2^2)^x = 2^{12}$

6. Apply the Power of a Power Rule

$(2^2)^x = 2^{2x}$

So the equation is:

$2^{2x} = 2^{12}$

7. Equate the Exponents

Since the bases are the same, equate the exponents:

$2x = 12$

8. Solve for $x$

Divide both sides by 2:

$x = \frac{12}{2} = 6$

Again, we find that $x = 6$. This alternative approach demonstrates that converting to a common base early can streamline the process and reduce complexity.

Common Mistakes to Avoid

When solving exponential equations, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure you arrive at the correct solution.

1. Incorrectly Applying Exponent Rules: Exponent rules are the foundation of solving exponential equations. Misapplying these rules can lead to significant errors. For instance, confusing $(a^m)^n$ with $a^{m+n}$ is a common mistake. Always double-check which rule applies in each situation.
2. Forgetting to Distribute Exponents: When raising a product or quotient to a power, remember to distribute the exponent to each factor. For example, $(ab)^n = a^n b^n$. Neglecting to do this can result in incorrect simplification.
3. Not Expressing Terms with a Common Base: One of the most effective strategies for solving exponential equations is to express all terms with a common base. Failing to do so can make the equation much harder to solve, if not impossible. Always look for opportunities to rewrite numbers as powers of the same base.
4. Making Arithmetic Errors: Simple arithmetic errors can derail your solution. Be careful when performing calculations, especially when dealing with fractions and negative exponents. It's always a good idea to double-check your work.
5. Ignoring the Order of Operations: Remember the order of operations (PEMDAS/BODMAS). Perform operations in the correct order to ensure accurate simplification. For example, simplify expressions inside parentheses before applying exponents.

By being mindful of these common mistakes, you can improve your accuracy and confidence in solving exponential equations.

Practice Problems

To solidify your understanding of solving exponential equations, let's work through a few practice problems. These examples will give you a chance to apply the concepts and techniques we've discussed.

Practice Problem 1: Solve for $x$ in the equation $3^{2x+1} = 27$.

• Solution:

  1. Rewrite 27 as a power of 3: $27 = 3^3$.
  2. The equation becomes $3^{2x+1} = 3^3$.
  3. Equate the exponents: $2x + 1 = 3$.
  4. Solve for $x$: $2x = 2$, so $x = 1$.

Practice Problem 2: Solve for $x$ in the equation $4^x = 8$.

• Solution:

  1. Rewrite both 4 and 8 as powers of 2: $4 = 2^2$ and $8 = 2^3$.
  2. The equation becomes $(2^2)^x = 2^3$.
  3. Apply the exponent rule $(a^m)^n = a^{mn}$: $2^{2x} = 2^3$.
  4. Equate the exponents: $2x = 3$.
  5. Solve for $x$: $x = \frac{3}{2}$.

Practice Problem 3: Solve for $x$ in the equation $5^{x^2 - 2x} = 125$.

• Solution:

  1. Rewrite 125 as a power of 5: $125 = 5^3$.
  2. The equation becomes $5^{x^2 - 2x} = 5^3$.
  3. Equate the exponents: $x^2 - 2x = 3$.
  4. Rearrange the equation to form a quadratic equation: $x^2 - 2x - 3 = 0$.
  5. Factor the quadratic equation: $(x - 3)(x + 1) = 0$.
  6. Solve for $x$: $x = 3$ or $x = -1$.

These practice problems illustrate the variety of exponential equations you might encounter and the different techniques you can use to solve them. Remember to always look for opportunities to simplify the equation and express terms with a common base.

Conclusion

In this guide, we've explored the process of solving the exponential equation $(\sqrt{\frac{\sqrt[4]{16^{12}}}{16^2}})^x=16^3$. We broke down the problem into manageable steps, applied exponent rules, and simplified the equation to find the value of $x$. We also discussed an alternative approach, common mistakes to avoid, and worked through practice problems to solidify your understanding.

Mastering exponential equations requires a strong grasp of exponent rules and algebraic manipulation. By following a systematic approach and practicing regularly, you can confidently solve these types of equations. Remember to always simplify the equation, express terms with a common base, and double-check your work to avoid common mistakes.

We hope this guide has provided you with the tools and knowledge you need to tackle exponential equations with confidence. Keep practicing, and you'll become proficient in no time!

For further exploration and more in-depth information on exponential equations, check out Khan Academy's Exponential Equations section. It's a fantastic resource for additional examples, practice problems, and video explanations.