Not equating the exponents

Oct 13, 2025 by windailys.com 18 views

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 type of exponential equation, where the base is the same on both sides of the equation. We will use the given equation as a case study to illustrate the steps involved in solving exponential equations.

The Given Equation

The given equation is:

${ 2^n = 2^9 \cdot 2^2 }$

Our goal is to find the value of n that satisfies this equation.

Step 1: Simplify the Right-Hand Side

To simplify the right-hand side of the equation, we can use the property of exponents that states:

${ a^m \cdot a^n = a^{m+n} }$

Applying this property to the given equation, we get:

${ 2^n = 2^{9+2} }$

${ 2^n = 2^{11} }$

Step 2: Equate the Exponents

Since the bases are the same on both sides of the equation, we can equate the exponents:

${ n = 11 }$

Conclusion

Based on the steps outlined above, we can conclude that the value of n that satisfies the given equation is:

${ n = 11 }$

Why is this the Correct Answer?

To understand why this is the correct answer, let's consider the properties of exponential functions. When we raise a number to a power, we are essentially multiplying that number by itself as many times as the exponent indicates. In this case, we have:

${ 2^{11} = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 }$

This is equivalent to multiplying 2 by itself 11 times. Therefore, the value of n that satisfies the given equation is indeed 11.

What if the Exponents were Different?

If the exponents were different, we would need to use a different approach to solve the equation. For example, if we had:

${ 2^n = 2^9 \cdot 2^3 }$

We could simplify the right-hand side using the property of exponents:

${ 2^n = 2^{9+3} }$

${ 2^n = 2^{12} }$

And then equate the exponents:

${ n = 12 }$

Conclusion

In conclusion, solving exponential equations requires a deep understanding of the underlying principles and the properties of exponents. By following the steps outlined above, we can solve equations of the form:

${ a^n = a^m \cdot a^k }$

Where a is a positive number and n, m, and k are integers.

Common Mistakes to Avoid

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

Not simplifying the right-hand side: Failing to simplify the right-hand side of the equation can lead to incorrect solutions.
Not equating the exponents: Failing to equate the exponents can lead to incorrect solutions.
Not considering the properties of exponents: Failing to consider the properties of exponents can lead to incorrect solutions.

Real-World Applications

Exponential equations have numerous real-world applications, including:

Finance: Exponential equations are used to model the growth of investments and the decay of assets.
Biology: Exponential equations are used to model the growth of populations and the decay of radioactive substances.
Computer Science: Exponential equations are used to model the time complexity of algorithms and the space complexity of data structures.

Conclusion

${ a^n = a^m \cdot a^k }$

Where a is a positive number and n, m, and k are integers. Exponential equations have numerous real-world applications, and understanding how to solve them is essential for success in a variety of fields.

Final Answer

The final answer is:

Q: What is an exponential equation?

A: An exponential equation is an equation in which the variable appears as an exponent. For example:

${ 2^n = 2^9 \cdot 2^2 }$

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to follow these steps:

Simplify the right-hand side of the equation using the properties of exponents.
Equate the exponents.
Solve for the variable.

Q: What are the properties of exponents?

A: The properties of exponents are:

Product of Powers: ${ a^m \cdot a^n = a^{m+n} }$
Power of a Power: ${ (a^m)^n = a^{m \cdot n} }$
Power of a Product: ${ (ab)^m = a^m \cdot b^m }$

Q: How do I simplify the right-hand side of an exponential equation?

A: To simplify the right-hand side of an exponential equation, you can use the properties of exponents. For example:

${ 2^n = 2^9 \cdot 2^2 }$

Using the product of powers property, we can simplify the right-hand side as follows:

${ 2^n = 2^{9+2} }$

${ 2^n = 2^{11} }$

Q: How do I equate the exponents?

A: To equate the exponents, you need to set the exponents on both sides of the equation equal to each other. For example:

${ 2^n = 2^{11} }$

Equating the exponents, we get:

${ n = 11 }$

Q: What if the exponents are different?

A: If the exponents are different, you need to use a different approach to solve the equation. For example:

${ 2^n = 2^9 \cdot 2^3 }$

Using the product of powers property, we can simplify the right-hand side as follows:

${ 2^n = 2^{9+3} }$

${ 2^n = 2^{12} }$

Equating the exponents, we get:

${ n = 12 }$

Q: Can I use logarithms to solve exponential equations?

A: Yes, you can use logarithms to solve exponential equations. For example:

${ 2^n = 2^9 \cdot 2^2 }$

Taking the logarithm of both sides, we get:

${ \log(2^n) = \log(2^9 \cdot 2^2) }$

Using the property of logarithms that states:

${ \log(a \cdot b) = \log(a) + \log(b) }$

We can simplify the right-hand side as follows:

${ \log(2^n) = \log(2^9) + \log(2^2) }$

${ \log(2^n) = 9 \log(2) + 2 \log(2) }$

${ \log(2^n) = 11 \log(2) }$

Equating the logarithms, we get:

${ n \log(2) = 11 \log(2) }$

Dividing both sides by ${\log(2)}$ , we get:

${ n = 11 }$

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

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

Not simplifying the right-hand side: Failing to simplify the right-hand side of the equation can lead to incorrect solutions.
Not equating the exponents: Failing to equate the exponents can lead to incorrect solutions.
Not considering the properties of exponents: Failing to consider the properties of exponents can lead to incorrect solutions.

Q: How do I apply exponential equations in real-world scenarios?

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

Finance: Exponential equations are used to model the growth of investments and the decay of assets.
Biology: Exponential equations are used to model the growth of populations and the decay of radioactive substances.
Computer Science: Exponential equations are used to model the time complexity of algorithms and the space complexity of data structures.

Conclusion

In conclusion, exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the underlying principles and the properties of exponents. By following the steps outlined above and avoiding common mistakes, you can solve exponential equations and apply them in real-world scenarios.