Using the Product Rule for Exponents

Oct 13, 2025 by windailys.com 123 views

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Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the underlying principles. In this article, we will focus on solving a specific exponential equation, which involves the product of two exponential expressions. Our goal is to find the value of the variable x that satisfies the given equation.

The Given Equation

The given equation is:

\left(6^{\frac{x}{2}}\right)\left(6^{\frac{x}{3}}\right)=6^8

This equation involves the product of two exponential expressions, both of which have a base of 6. The exponents of the two expressions are $\frac{x}{2}$ and $\frac{x}{3}$ , respectively.

Using the Product Rule for Exponents

To solve this equation, we can use the product rule for exponents, which states that when multiplying two exponential expressions with the same base, we can add the exponents. In this case, the product rule can be applied as follows:

\left(6^{\frac{x}{2}}\right)\left(6^{\frac{x}{3}}\right)=6^{\frac{x}{2}+\frac{x}{3}}

Simplifying the Exponent

Now, we can simplify the exponent by finding a common denominator. The common denominator of 2 and 3 is 6, so we can rewrite the exponent as follows:

\frac{x}{2}+\frac{x}{3}=\frac{3x}{6}+\frac{2x}{6}=\frac{5x}{6}

Equating the Exponents

Now that we have simplified the exponent, we can equate it to the exponent on the right-hand side of the equation:

\frac{5x}{6}=8

Solving for x

To solve for x, we can multiply both sides of the equation by $\frac{6}{5}$ :

x=\frac{6}{5}\cdot8=\frac{48}{5}

Conclusion

In this article, we have solved a specific exponential equation that involved the product of two exponential expressions. We used the product rule for exponents to simplify the equation and then equated the exponents to solve for x. The final answer is $\boxed{\frac{48}{5}}$ .

Tips and Tricks

When solving exponential equations, it's essential to use the product rule for exponents to simplify the equation.
Make sure to find a common denominator when adding or subtracting fractions with different denominators.
When equating exponents, make sure to use the same base and exponent on both sides of the equation.

Real-World Applications

Exponential equations have numerous real-world applications in fields such as finance, economics, and computer science. For example, exponential growth and decay are used to model population growth, chemical reactions, and financial investments.

Practice Problems

Here are a few practice problems to help you reinforce your understanding of solving exponential equations:

Solve the equation: $\left(2^{\frac{x}{4}}\right)\left(2^{\frac{x}{6}}\right)=2^{12}$
Solve the equation: $\left(3^{\frac{x}{2}}\right)\left(3^{\frac{x}{5}}\right)=3^{15}$
Solve the equation: $\left(4^{\frac{x}{3}}\right)\left(4^{\frac{x}{7}}\right)=4^{21}$

Conclusion

Solving exponential equations requires a deep understanding of the underlying principles and a step-by-step approach. By using the product rule for exponents and simplifying the exponent, we can solve for x and find the solution to the equation. We hope that this article has provided you with a comprehensive guide to solving exponential equations and has helped you to develop your problem-solving skills.

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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. Exponential equations can be written in the form $a^x = b$ , where $a$ is the base, $x$ is the exponent, and $b$ is the result.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable $x$ . This can be done by using the properties of exponents, such as the product rule and the power rule. You can also use logarithms to solve exponential equations.

Q: What is the product rule for exponents?

A: The product rule for exponents states that when multiplying two exponential expressions with the same base, you can add the exponents. For example, $a^m \cdot a^n = a^{m+n}$ .

Q: How do I simplify an exponential expression?

A: To simplify an exponential expression, you can use the properties of exponents, such as the product rule and the power rule. You can also use logarithms to simplify exponential expressions.

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. Logarithmic equations can be written in the form $\log_a x = y$ , where $a$ is the base, $x$ is the result, and $y$ is the exponent.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to isolate the variable $y$ . This can be done by using the properties of logarithms, such as the product rule and the power rule. You can also use exponential functions to solve logarithmic equations.

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 rule for exponents
Not simplifying the exponential expression
Not using logarithms to solve the equation
Not checking the solution to make sure it is valid

Q: How do I check my solution to an exponential equation?

A: To check your solution to an exponential equation, you need to plug the solution back into the original equation and make sure it is true. This can be done by using a calculator or by simplifying the equation.

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

A: Exponential equations have numerous real-world applications in fields such as finance, economics, and computer science. For example, exponential growth and decay are used to model population growth, chemical reactions, and financial investments.

Q: How do I use exponential equations in real-world applications?

A: To use exponential equations in real-world applications, you need to understand the underlying principles and be able to apply them to real-world problems. This can be done by using the properties of exponents and logarithms, and by using calculators or computer software to solve the equations.

Q: What are some common types of exponential equations?

A: Some common types of exponential equations include:

Linear exponential equations: $a^x = b$
Quadratic exponential equations: $a^{x^2} = b$
Exponential equations with a variable base: $a^x = b$ , where $a$ is a variable

Q: How do I solve exponential equations with a variable base?

A: To solve exponential equations with a variable base, you need to use the properties of exponents and logarithms. You can also use calculators or computer software to solve the equations.

Q: What are some common mistakes to avoid when solving exponential equations with a variable base?

A: Some common mistakes to avoid when solving exponential equations with a variable base include:

Not using the product rule for exponents
Not simplifying the exponential expression
Not using logarithms to solve the equation
Not checking the solution to make sure it is valid

Q: How do I check my solution to an exponential equation with a variable base?

A: To check your solution to an exponential equation with a variable base, you need to plug the solution back into the original equation and make sure it is true. This can be done by using a calculator or by simplifying the equation.

Q: What are some real-world applications of exponential equations with a variable base?

A: Exponential equations with a variable base have numerous real-world applications in fields such as finance, economics, and computer science. For example, exponential growth and decay are used to model population growth, chemical reactions, and financial investments.

Q: How do I use exponential equations with a variable base in real-world applications?

A: To use exponential equations with a variable base in real-world applications, you need to understand the underlying principles and be able to apply them to real-world problems. This can be done by using the properties of exponents and logarithms, and by using calculators or computer software to solve the equations.