Solving Exponential Equations $9^{x^2}=9$ Algebraically

by ADMIN 56 views

In the realm of mathematics, exponential equations hold a significant position, often presenting intriguing challenges that demand a blend of algebraic manipulation and a solid understanding of exponential properties. This article delves into the process of solving the exponential equation 9x2=99^{x^2} = 9 algebraically, providing a step-by-step guide to arrive at the solution. We will also explore the underlying concepts and techniques involved, ensuring a comprehensive understanding of this mathematical domain.

Understanding Exponential Equations

Before we embark on the solution process, it's crucial to grasp the essence of exponential equations. An exponential equation is an equation in which the variable appears in the exponent. These equations arise in various contexts, including population growth, radioactive decay, and compound interest calculations. The key to solving exponential equations lies in manipulating the equation to isolate the variable, often by employing the properties of exponents and logarithms.

Solving the Equation 9x2=99^{x^2} = 9

Now, let's tackle the equation at hand: 9x2=99^{x^2} = 9. Our objective is to determine the value(s) of x that satisfy this equation. We can achieve this by following a systematic algebraic approach:

1. Express Both Sides with the Same Base

The cornerstone of solving many exponential equations is to express both sides of the equation with the same base. In our case, we observe that both 9 and 9 can be expressed as powers of 3. Specifically, 9=329 = 3^2. Substituting this into our equation, we get:

(32)x2=32(3^2)^{x^2} = 3^2

2. Apply the Power of a Power Rule

The power of a power rule states that (am)n=am∗n(a^m)^n = a^{m*n}. Applying this rule to the left side of our equation, we obtain:

32x2=323^{2x^2} = 3^2

3. Equate the Exponents

Now that both sides of the equation have the same base (3), we can equate the exponents. This is a fundamental property of exponential equations: if am=ana^m = a^n, then m = n (provided a is not 0, 1, or -1). Equating the exponents in our equation gives us:

2x2=22x^2 = 2

4. Solve for x2x^2

To isolate x2x^2, we divide both sides of the equation by 2:

x2=1x^2 = 1

5. Solve for xx

To find the values of x, we take the square root of both sides of the equation. Remember that when taking the square root, we must consider both positive and negative roots:

x=±√1x = ±√1

This yields two solutions:

x=1x = 1 or x=−1x = -1

Therefore, the solutions to the equation 9x2=99^{x^2} = 9 are x = 1 and x = -1.

Verification

It's always a good practice to verify our solutions by substituting them back into the original equation. Let's check if our solutions hold true:

For x = 1:

9(1)2=91=99^{(1)^2} = 9^1 = 9

This confirms that x = 1 is indeed a solution.

For x = -1:

9(−1)2=91=99^{(-1)^2} = 9^1 = 9

This also verifies that x = -1 is a solution.

Approximate the result(s) to three decimal places, if necessary

In this case, the solutions 1 and -1 are already exact integers, so approximating them to three decimal places would simply yield 1.000 and -1.000, respectively. While these are technically approximations, they don't provide any additional information since the original solutions are already precise.

General Strategies for Solving Exponential Equations

While the equation we tackled was relatively straightforward, many exponential equations require more intricate techniques. Here are some general strategies to keep in mind:

  • Express with the Same Base: As demonstrated in our example, expressing both sides of the equation with the same base is often the first crucial step.
  • Use Logarithms: When it's not feasible to express both sides with the same base, logarithms come to the rescue. Logarithms are the inverse functions of exponentials, and they allow us to bring the exponent down as a coefficient.
  • Substitution: In some cases, substituting a new variable for a complex exponential expression can simplify the equation.
  • Quadratic Form: Certain exponential equations can be transformed into quadratic equations through substitution, allowing us to use the quadratic formula or factoring techniques.

Common Mistakes to Avoid

When solving exponential equations, it's essential to be wary of common pitfalls:

  • Incorrectly Applying Exponent Rules: A solid grasp of exponent rules is crucial. Mistakes in applying these rules can lead to erroneous results.
  • Ignoring Negative Roots: When taking square roots or other even roots, remember to consider both positive and negative solutions.
  • Dividing by Zero: Avoid dividing both sides of an equation by an expression that could be zero.
  • Forgetting to Verify Solutions: Always verify your solutions by substituting them back into the original equation.

Conclusion

Solving exponential equations is a fundamental skill in mathematics, with applications spanning various fields. By understanding the properties of exponents and employing strategic algebraic techniques, we can effectively tackle these equations. The equation 9x2=99^{x^2} = 9 served as a valuable example, illustrating the power of expressing both sides with the same base and equating exponents. As you continue your mathematical journey, remember to practice and refine your skills in solving exponential equations, as they will undoubtedly prove to be invaluable tools.

By mastering the techniques discussed in this article, you'll be well-equipped to solve a wide range of exponential equations, gaining a deeper appreciation for the elegance and power of mathematics.

In summary, the solutions to the equation 9x2=99^{x^2} = 9 are 1, -1.