Using the Power of a Power Property

Oct 13, 2025 by windailys.com 88 views

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of exponents. In this article, we will explore how to solve exponential equations, with a focus on the given equation: $2^6 \times 2^{12} = \left(2^x\right)^2$ . We will break down the solution into manageable steps, using the properties of exponents to simplify the equation and isolate the variable.

Understanding Exponents

Before we dive into the solution, let's review the basics of exponents. An exponent is a small number that is raised to a power, indicating how many times a base number is multiplied by itself. For example, $2^3$ means $2$ multiplied by itself $3$ times, or $2 \times 2 \times 2 = 8$ . Exponents can also be negative, which means the base number is divided by itself a certain number of times. For example, $2^{-3}$ means $2$ divided by itself $3$ times, or $\frac{1}{2 \times 2 \times 2} = \frac{1}{8}$ .

Simplifying the Equation

Now that we have a solid understanding of exponents, let's simplify the given equation: $2^6 \times 2^{12} = \left(2^x\right)^2$ . To simplify the equation, we can use the property of exponents that states when we multiply two numbers with the same base, we can add their exponents. In this case, we have:

2^6 \times 2^{12} = 2^{6+12} = 2^{18}

So, the equation becomes:

2^{18} = \left(2^x\right)^2

Using the Power of a Power Property

The next step is to use the power of a power property, which states that when we raise a power to a power, we can multiply the exponents. In this case, we have:

\left(2^x\right)^2 = 2^{2x}

So, the equation becomes:

2^{18} = 2^{2x}

Equating the Exponents

Now that we have simplified the equation, we can equate the exponents. Since the bases are the same, we can set the exponents equal to each other:

18 = 2x

Solving for x

To solve for x, we can divide both sides of the equation by 2:

x = \frac{18}{2}

x = 9

Conclusion

In this article, we have solved the exponential equation $2^6 \times 2^{12} = \left(2^x\right)^2$ using the properties of exponents. We simplified the equation by adding the exponents, used the power of a power property to raise the power to a power, and equated the exponents to solve for x. The final answer is x = 9.

Real-World Applications

Exponential equations have many real-world applications, including:

Finance: Exponential equations are used to calculate compound interest, which is the interest earned on both the principal amount and any accrued interest over time.
Science: Exponential equations are used to model population growth, chemical reactions, and other phenomena that involve exponential change.
Engineering: Exponential equations are used to design and optimize systems, such as electronic circuits and mechanical systems.

Common Mistakes to Avoid

When solving exponential equations, there are several common mistakes to avoid:

Not using the properties of exponents: Failing to use the properties of exponents, such as the power of a power property, can lead to incorrect solutions.
Not equating the exponents: Failing to equate the exponents can lead to incorrect solutions.
Not checking the solution: Failing to check the solution can lead to incorrect answers.

Practice Problems

To practice solving exponential equations, try the following problems:

$3^4 \times 3^8 = \left(3^x\right)^3$
$2^5 \times 2^{10} = \left(2^x\right)^2$
$4^3 \times 4^6 = \left(4^x\right)^2$

Conclusion

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponent, which is a small number that is raised to a power, indicating how many times a base number is multiplied by itself.

Q: What are the properties of exponents?

A: The properties of exponents include:

Power of a power property: When we raise a power to a power, we can multiply the exponents.
Product of powers property: When we multiply two numbers with the same base, we can add their exponents.
Quotient of powers property: When we divide two numbers with the same base, we can subtract their exponents.

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, you can use the properties of exponents to combine the terms. For example, if you have $2^6 \times 2^{12}$ , you can add the exponents to get $2^{6+12} = 2^{18}$ .

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use the properties of exponents to isolate the variable. For example, if you have $2^{18} = \left(2^x\right)^2$ , you can use the power of a power property to raise the power to a power, and then equate the exponents to solve for x.

Q: What are some common mistakes to avoid when solving exponential equations?

A: Some common mistakes to avoid when solving exponential equations include:

Not using the properties of exponents: Failing to use the properties of exponents can lead to incorrect solutions.
Not equating the exponents: Failing to equate the exponents can lead to incorrect solutions.
Not checking the solution: Failing to check the solution can lead to incorrect answers.

Q: How do I check my solution?

A: To check your solution, you can plug the value of x back into the original equation and simplify. If the equation is true, then your solution is correct.

Q: What are some real-world applications of exponential equations?

A: Exponential equations have many real-world applications, including:

Finance: Exponential equations are used to calculate compound interest, which is the interest earned on both the principal amount and any accrued interest over time.
Science: Exponential equations are used to model population growth, chemical reactions, and other phenomena that involve exponential change.
Engineering: Exponential equations are used to design and optimize systems, such as electronic circuits and mechanical systems.

Q: How can I practice solving exponential equations?

A: You can practice solving exponential equations by working through problems and exercises, such as:

Simplifying exponential expressions: Simplify expressions like $2^6 \times 2^{12}$ .
Solving exponential equations: Solve equations like $2^{18} = \left(2^x\right)^2$ .
Modeling real-world problems: Use exponential equations to model real-world problems, such as population growth or compound interest.

Q: What are some resources for learning more about exponential equations?

A: Some resources for learning more about exponential equations include:

Textbooks: Textbooks on algebra and mathematics often cover exponential equations in detail.
Online resources: Websites like Khan Academy, Mathway, and Wolfram Alpha offer interactive lessons and exercises on exponential equations.
Tutorials and videos: YouTube channels like 3Blue1Brown, Crash Course, and Vi Hart offer video tutorials and explanations on exponential equations.

Conclusion

In conclusion, solving exponential equations requires a deep understanding of the properties of exponents. By simplifying the equation, using the power of a power property, and equating the exponents, we can solve for x. The final answer is x = 9. Exponential equations have many real-world applications, and by avoiding common mistakes and practicing with problems, we can become proficient in solving them.