Using the Product of Powers Property

Oct 13, 2025 by magicalceylontours.com 136 views

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Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of exponents and logarithms. In this article, we will focus on solving the equation $5^3 \cdot 5^x = 5^{12}$ , which is a classic example of an exponential equation. We will break down the solution into manageable steps, making it easy to understand and follow.

Understanding Exponents

Before we dive into the solution, let's quickly review the concept 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, $5^3$ means $5$ multiplied by itself $3$ times, which equals $125$ . Exponents can also be negative, which means the base number is divided by itself a certain number of times.

The Equation $5^3 \cdot 5^x = 5^{12}$

Now that we have a basic understanding of exponents, let's examine the given equation: $5^3 \cdot 5^x = 5^{12}$ . This equation involves two exponential expressions, $5^3$ and $5^x$ , which are multiplied together to equal $5^{12}$ . Our goal is to solve for the variable $x$ .

Using the Product of Powers Property

To solve this equation, we can use the product of powers property, which states that when multiplying two exponential expressions with the same base, we can add their exponents. In this case, we have:

5^3 \cdot 5^x = 5^{3+x}

Now, we can equate this expression to $5^{12}$ :

5^{3+x} = 5^{12}

Equating Exponents

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

3+x = 12

Solving for $x$

Now, we can solve for $x$ by subtracting $3$ from both sides of the equation:

x = 12-3

x = 9

Therefore, the value of $x$ that satisfies the equation $5^3 \cdot 5^x = 5^{12}$ is $x = 9$ .

Conclusion

Solving exponential equations requires a deep understanding of exponents and logarithms. By using the product of powers property and equating exponents, we can solve for the variable $x$ . In this article, we solved the equation $5^3 \cdot 5^x = 5^{12}$ , which resulted in the value of $x = 9$ . We hope this article has provided a clear and concise guide to solving exponential equations.

Real-World Applications

Exponential equations have numerous real-world applications, including:

Finance: Exponential equations are used to calculate compound interest and investment growth.
Biology: Exponential equations are used to model population growth and decay.
Computer Science: Exponential equations are used to analyze the time complexity of algorithms.

Common Mistakes to Avoid

When solving exponential equations, it's essential to avoid common mistakes, including:

Not using the product of powers property: Failing to use this property can lead to incorrect solutions.
Not equating exponents: Failing to equate exponents can lead to incorrect solutions.
Not checking for extraneous solutions: Failing to check for extraneous solutions can lead to incorrect solutions.

Tips and Tricks

Here are some tips and tricks to help you solve exponential equations:

Use the product of powers property: This property can help you simplify exponential expressions and solve equations.
Equating exponents: Equating exponents is a crucial step in solving exponential equations.
Check for extraneous solutions: Always check for extraneous solutions to ensure that your solution is correct.

Practice Problems

Here are some practice problems to help you practice solving exponential equations:

Problem 1: Solve the equation $2^x \cdot 2^3 = 2^{12}$ .
Problem 2: Solve the equation $3^x \cdot 3^2 = 3^{15}$ .
Problem 3: Solve the equation $4^x \cdot 4^4 = 4^{20}$ .

Conclusion

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Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential expression, which is a number raised to a power. For example, $5^3$ is an exponential expression, where $5$ is the base and $3$ is the exponent.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to use the product of powers property, which states that when multiplying two exponential expressions with the same base, you can add their exponents. You also need to equate the exponents and solve for the variable.

Q: What is the product of powers property?

A: The product of powers property is a mathematical rule that states that when multiplying two exponential expressions with the same base, you can add their exponents. For example, $5^3 \cdot 5^x = 5^{3+x}$ .

Q: How do I use the product of powers property?

A: To use the product of powers property, you need to identify the bases and exponents in the equation. Then, you can add the exponents and equate the result to the original equation.

Q: What is an extraneous solution?

A: An extraneous solution is a solution that is not valid or is not a solution to the original equation. For example, if you solve the equation $5^3 \cdot 5^x = 5^{12}$ and get $x = 9$ , but then check the solution and find that it is not valid, then $x = 9$ is an extraneous solution.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you need to plug the solution back into the original equation and check if it is true. If the solution is not true, then it is an extraneous solution.

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 product of powers property
Not equating exponents
Not checking for extraneous solutions

Q: How do I practice solving exponential equations?

A: To practice solving exponential equations, you can try solving problems on your own or use online resources such as practice tests or worksheets.

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

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

Finance: Exponential equations are used to calculate compound interest and investment growth.
Biology: Exponential equations are used to model population growth and decay.
Computer Science: Exponential equations are used to analyze the time complexity of algorithms.

Q: How do I know if an equation is an exponential equation?

A: An equation is an exponential equation if it involves an exponential expression, which is a number raised to a power. For example, $5^3$ is an exponential expression.

Q: Can I use logarithms to solve exponential equations?

A: Yes, you can use logarithms to solve exponential equations. Logarithms can help you simplify exponential expressions and solve equations.

Q: What is the difference between an exponential equation and a logarithmic equation?

A: An exponential equation is an equation that involves an exponential expression, while a logarithmic equation is an equation that involves a logarithmic expression. For example, $5^3 = 125$ is an exponential equation, while $\log_5 125 = 3$ is a logarithmic equation.

Q: Can I use a calculator to solve exponential equations?

A: Yes, you can use a calculator to solve exponential equations. However, it's always a good idea to check your solution by plugging it back into the original equation.

Q: How do I know if a solution is valid or not?

A: To know if a solution is valid or not, you need to plug it back into the original equation and check if it is true. If the solution is not true, then it is not a valid solution.