Not applying the properties of exponents correctly

Oct 13, 2025 by leolavanderia.com.br 96 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 the process of solving exponential equations, with a focus on the equation $2^{4x} = 8^{x-3}$ . We will examine the properties of exponents, apply the rules of equality, and use algebraic manipulations to simplify the equation.

Understanding Exponents

Exponents are a shorthand way of writing repeated multiplication. For example, $2^3$ can be written as $2 \times 2 \times 2$ , or $8$ . Exponents can also be used to represent negative numbers, such as $2^{-3}$ , which is equal to $\frac{1}{2^3}$ or $\frac{1}{8}$ .

Properties of Exponents

There are several properties of exponents that are essential to solving exponential equations. These properties include:

Product of Powers: When multiplying two powers with the same base, we can add the exponents. For example, $2^3 \times 2^4 = 2^{3+4} = 2^7$ .
Power of a Power: When raising a power to another power, we can multiply the exponents. For example, $(2^3)^4 = 2^{3 \times 4} = 2^{12}$ .
Quotient of Powers: When dividing two powers with the same base, we can subtract the exponents. For example, $\frac{2^5}{2^3} = 2^{5-3} = 2^2$ .

Solving the Equation

Now that we have a solid understanding of the properties of exponents, we can apply them to solve the equation $2^{4x} = 8^{x-3}$ .

First, we can rewrite the equation using the fact that $8 = 2^3$ . This gives us:

2^{4x} = (2^3)^{x-3}

Using the property of exponents that $(a^m)^n = a^{m \times n}$ , we can rewrite the equation as:

2^{4x} = 2^{3(x-3)}

Now, we can use the property of exponents that $a^m \times a^n = a^{m+n}$ to simplify the equation. This gives us:

2^{4x} = 2^{3x-9}

Comparing the Options

Now that we have simplified the equation, we can compare it to the options provided. We can see that option D, $2^{4x} = 2^{3x-9}$ , is equivalent to the simplified equation.

Conclusion

In this article, we have explored the process of solving exponential equations, with a focus on the equation $2^{4x} = 8^{x-3}$ . We have applied the properties of exponents, used algebraic manipulations to simplify the equation, and compared the result to the options provided. By following these steps, we can confidently solve exponential equations and understand the underlying mathematics.

Final Answer

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, $2^3$ is an exponential expression, where 2 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 apply the properties of exponents, such as the product of powers, power of a power, and quotient of powers. You also need to use algebraic manipulations to simplify the equation.

Q: What are the properties of exponents?

A: The properties of exponents are:

Product of Powers: When multiplying two powers with the same base, we can add the exponents. For example, $2^3 \times 2^4 = 2^{3+4} = 2^7$ .
Power of a Power: When raising a power to another power, we can multiply the exponents. For example, $(2^3)^4 = 2^{3 \times 4} = 2^{12}$ .
Quotient of Powers: When dividing two powers with the same base, we can subtract the exponents. For example, $\frac{2^5}{2^3} = 2^{5-3} = 2^2$ .

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, you need to apply the properties of exponents and use algebraic manipulations to rewrite the equation in a simpler form.

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

A: An exponential equation involves an exponential expression, while a logarithmic equation involves a logarithmic expression. For example, $2^3 = 8$ is an exponential equation, while $\log_2 8 = 3$ is a logarithmic equation.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to apply the properties of logarithms, such as the product of logarithms, power of a logarithm, and quotient of logarithms. You also need to use algebraic manipulations to simplify the equation.

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

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

Not applying the properties of exponents correctly
Not simplifying the equation properly
Not checking the solution for extraneous solutions

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 verify that it is true. If the solution is not true, then it is an extraneous solution.

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

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

Population growth: Exponential equations can be used to model population growth, where the population grows at a rate proportional to the current population.
Financial modeling: Exponential equations can be used to model financial growth, where the value of an investment grows at a rate proportional to the current value.
Science and engineering: Exponential equations can be used to model many scientific and engineering phenomena, such as radioactive decay, chemical reactions, and electrical circuits.