(ii) Solve For { X $}$ In The Equation: ${ 7^3 \times 7^{2x} = 7^{13} }$

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 focus on solving exponential equations with the same base, specifically the equation $7^3 \times 7^{2x} = 7^{13}$. We will use the properties of exponents to simplify the equation and solve for the variable $x$.

Understanding Exponents

Before we dive into solving the equation, let's review the properties of exponents. The exponent of a number is the power to which the number is raised. For example, in the expression $7^3$, the exponent is 3, and the base is 7. When we multiply two numbers with the same base, we add their exponents. For example, $7^3 \times 7^2 = 7^{3+2} = 7^5$.

Simplifying the Equation

Now that we have reviewed the properties of exponents, let's simplify the equation $7^3 \times 7^{2x} = 7^{13}$. We can start by using the property of multiplication with the same base to combine the left-hand side of the equation:

$$7^3 \times 7^{2x} = 7^{3+2x}$$

Now we have the equation $7^{3+2x} = 7^{13}$. Since the bases are the same, we can equate the exponents:

$$3+2x = 13$$

Solving for $x$

Now that we have a linear equation, we can solve for $x$. We can start by subtracting 3 from both sides of the equation:

$$2x = 10$$

Next, we can divide both sides of the equation by 2:

$$x = 5$$

Therefore, the solution to the equation $7^3 \times 7^{2x} = 7^{13}$ is $x = 5$.

Conclusion

Solving exponential equations with the same base requires a deep understanding of the properties of exponents. By using the properties of multiplication and addition with the same base, we can simplify the equation and solve for the variable $x$. In this article, we have solved the equation $7^3 \times 7^{2x} = 7^{13}$ and found that the solution is $x = 5$.

Example Use Cases

Exponential equations with the same base have many real-world applications. For example, in finance, exponential growth is used to model the growth of investments. In biology, exponential growth is used to model the growth of populations. In computer science, exponential growth is used to model the growth of data.

Tips and Tricks

When solving exponential equations with the same base, it's essential to remember the properties of exponents. Specifically, when multiplying two numbers with the same base, you add their exponents. When dividing two numbers with the same base, you subtract their exponents. By remembering these properties, you can simplify the equation and solve for the variable $x$.

Common Mistakes

When solving exponential equations with the same base, it's easy to make mistakes. One common mistake is to forget to use the properties of exponents. For example, in the equation $7^3 \times 7^{2x} = 7^{13}$, it's easy to forget to add the exponents when multiplying the two numbers with the same base. By remembering the properties of exponents, you can avoid making this mistake and solve the equation correctly.

Conclusion

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, the equation $7^3 \times 7^{2x} = 7^{13}$ is an exponential equation.

Q: What is the base of an exponential equation?

A: The base of an exponential equation is the number that is raised to a power. In the equation $7^3 \times 7^{2x} = 7^{13}$, the base is 7.

Q: What is the exponent of an exponential equation?

A: The exponent of an exponential equation is the power to which the base is raised. In the equation $7^3 \times 7^{2x} = 7^{13}$, the exponent is 3 and $2x$.

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, you can use the properties of exponents. Specifically, when multiplying two numbers with the same base, you add their exponents. When dividing two numbers with the same base, you subtract their exponents.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use the properties of exponents to simplify the equation and then solve for the variable. For example, in the equation $7^3 \times 7^{2x} = 7^{13}$, you can simplify the equation by adding the exponents and then 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:

• Forgetting to use the properties of exponents
• Not simplifying the equation before solving for the variable
• Not checking the solution to make sure it is correct

Q: How do I apply exponential equations to real-world problems?

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

• Modeling the growth of investments in finance
• Modeling the growth of populations in biology
• Modeling the growth of data in computer science

Q: What are some tips and tricks for solving exponential equations?

A: Some tips and tricks for solving exponential equations include:

• Remembering the properties of exponents
• Simplifying the equation before solving for the variable
• Checking the solution to make sure it is correct

Q: How do I practice solving exponential equations?

A: You can practice solving exponential equations by:

• Working through examples and exercises in a textbook or online resource
• Creating your own problems and solutions
• Joining a study group or working with a tutor to practice solving exponential equations.

Conclusion

Solving exponential equations requires a deep understanding of the properties of exponents and the ability to simplify and solve equations. By remembering the properties of exponents, simplifying the equation, and checking the solution, you can solve exponential equations and apply them to real-world problems.