To Solve $49^{3x} = 343^{2x+1}$, Write Each Side Of The Equation In Terms Of The Same Base.

by ADMIN 94 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 focus on solving the exponential equation $49^{3x} = 343^{2x+1}$ by rewriting each side of the equation in terms of the same base.

Understanding the Properties of Exponents

Before we dive into solving the equation, it's essential to understand the properties of exponents. The exponent of a number is the power to which the number is raised. For example, in the expression $a^b$, $a$ is the base and $b$ is the exponent.

One of the most important properties of exponents is the rule of multiplication, which states that when we multiply two numbers with the same base, we add their exponents. For example, $a^b \cdot a^c = a^{b+c}$.

Another important property of exponents is the rule of division, which states that when we divide two numbers with the same base, we subtract their exponents. For example, $\frac{ab}{ac} = a^{b-c}$.

Rewriting the Equation in Terms of the Same Base

Now that we have a good understanding of the properties of exponents, let's rewrite the equation $49^{3x} = 343^{2x+1}$ in terms of the same base.

We know that $49 = 7^2$ and $343 = 7^3$. Therefore, we can rewrite the equation as follows:

(72)3x=(73)2x+1(7^2)^{3x} = (7^3)^{2x+1}

Using the rule of multiplication, we can simplify the left-hand side of the equation as follows:

72â‹…3x=73â‹…(2x+1)7^{2 \cdot 3x} = 7^{3 \cdot (2x+1)}

Simplifying further, we get:

76x=76x+37^{6x} = 7^{6x+3}

Setting the Exponents Equal to Each Other

Now that we have rewritten the equation in terms of the same base, we can set the exponents equal to each other. This gives us the following equation:

6x=6x+36x = 6x+3

Solving for x

Now that we have set the exponents equal to each other, we can solve for $x$. Subtracting $6x$ from both sides of the equation, we get:

0=30 = 3

This is a contradiction, which means that the original equation $49^{3x} = 343^{2x+1}$ has no solution.

Conclusion

In this article, we have shown how to solve the exponential equation $49^{3x} = 343^{2x+1}$ by rewriting each side of the equation in terms of the same base. We have also demonstrated the importance of understanding the properties of exponents in solving exponential equations.

By following the steps outlined in this article, you should be able to solve exponential equations with ease. Remember to always rewrite the equation in terms of the same base and set the exponents equal to each other to solve for the variable.

Frequently Asked Questions

  • What is the base of the exponential equation? The base of the exponential equation is 7.
  • How do I rewrite the equation in terms of the same base? To rewrite the equation in terms of the same base, you need to express both sides of the equation in terms of the same base. In this case, we used the fact that $49 = 7^2$ and $343 = 7^3$ to rewrite the equation.
  • How do I set the exponents equal to each other? To set the exponents equal to each other, you need to equate the exponents of the two sides of the equation. In this case, we set $6x = 6x+3$.

Further Reading

  • Exponential Equations Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of exponents. In this article, we have shown how to solve the exponential equation $49^{3x} = 343^{2x+1}$ by rewriting each side of the equation in terms of the same base.
  • Properties of Exponents The properties of exponents are a crucial concept in mathematics, and understanding them is essential for solving exponential equations. In this article, we have demonstrated the importance of understanding the properties of exponents in solving exponential equations.
  • Solving Exponential Equations Solving exponential equations requires a deep understanding of the properties of exponents. In this article, we have shown how to solve the exponential equation $49^{3x} = 343^{2x+1}$ by rewriting each side of the equation in terms of the same base.

=====================================

Frequently Asked Questions

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.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to rewrite the equation in terms of the same base and set the exponents equal to each other. For example, to solve the equation $49^{3x} = 343^{2x+1}$, you would rewrite it as $(72){3x} = (73){2x+1}$ and then set the exponents equal to each other.

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. For example, in the equation $2^3$, the base is 2.

Q: How do I rewrite an exponential equation in terms of the same base?

A: To rewrite an exponential equation in terms of the same base, you need to express both sides of the equation in terms of the same base. For example, to rewrite the equation $49^{3x} = 343^{2x+1}$, you would use the fact that $49 = 7^2$ and $343 = 7^3$ to rewrite it as $(72){3x} = (73){2x+1}$.

Q: How do I set the exponents equal to each other?

A: To set the exponents equal to each other, you need to equate the exponents of the two sides of the equation. For example, to solve the equation $(72){3x} = (73){2x+1}$, you would set $2 \cdot 3x = 3 \cdot (2x+1)$.

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, $2^3 = 8$ 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 rewrite the equation in exponential form and then solve for the variable. For example, to solve the equation $\log_2 8 = 3$, you would rewrite it as $2^3 = 8$ and then solve for the variable.

Q: What is the relationship between exponential and logarithmic equations?

A: Exponential and logarithmic equations are inverse operations. For example, $2^3 = 8$ is an exponential equation, while $\log_2 8 = 3$ is a logarithmic equation. This means that if you have an exponential equation, you can rewrite it as a logarithmic equation and vice versa.

Q: How do I use a calculator to solve an exponential equation?

A: To use a calculator to solve an exponential equation, you need to enter the equation in the calculator and then use the exponent key to solve for the variable. For example, to solve the equation $2^3 = 8$, you would enter the equation in the calculator and then use the exponent key to solve for the variable.

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

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

  • Not rewriting the equation in terms of the same base
  • Not setting the exponents equal to each other
  • Not using the correct exponent key on the calculator
  • Not checking the solution for extraneous solutions

Q: How do I check my solution for extraneous solutions?

A: To check your solution for extraneous solutions, you need to plug the solution back into the original equation and check if it is true. For example, to check the solution $x = 3$ to the equation $2^3 = 8$, you would plug $x = 3$ back into the equation and check if it is true.

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

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

  • Modeling population growth
  • Modeling chemical reactions
  • Modeling financial growth
  • Modeling electrical circuits

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

A: To use exponential equations to model real-world problems, you need to identify the variables and the relationships between them. For example, to model population growth, you would use an exponential equation to model the growth of the population over time.

Q: What are some common types of exponential equations?

A: Some common types of exponential equations include:

  • Linear exponential equations
  • Quadratic exponential equations
  • Polynomial exponential equations
  • Rational exponential equations

Q: How do I solve linear exponential equations?

A: To solve linear exponential equations, you need to rewrite the equation in terms of the same base and then set the exponents equal to each other. For example, to solve the equation $2^3x = 8$, you would rewrite it as $(23)x = 2^3$ and then set the exponents equal to each other.

Q: How do I solve quadratic exponential equations?

A: To solve quadratic exponential equations, you need to rewrite the equation in terms of the same base and then set the exponents equal to each other. For example, to solve the equation $2{3x2} = 8$, you would rewrite it as $(23){x^2} = 2^3$ and then set the exponents equal to each other.

Q: How do I solve polynomial exponential equations?

A: To solve polynomial exponential equations, you need to rewrite the equation in terms of the same base and then set the exponents equal to each other. For example, to solve the equation $2{3x3} = 8$, you would rewrite it as $(23){x^3} = 2^3$ and then set the exponents equal to each other.

Q: How do I solve rational exponential equations?

A: To solve rational exponential equations, you need to rewrite the equation in terms of the same base and then set the exponents equal to each other. For example, to solve the equation $2{3x3} = 8$, you would rewrite it as $(23){x^3} = 2^3$ and then set the exponents equal to each other.

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

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

  • Not rewriting the equation in terms of the same base
  • Not setting the exponents equal to each other
  • Not using the correct exponent key on the calculator
  • Not checking the solution for extraneous solutions

Q: How do I use a calculator to solve exponential equations?

A: To use a calculator to solve exponential equations, you need to enter the equation in the calculator and then use the exponent key to solve for the variable. For example, to solve the equation $2^3 = 8$, you would enter the equation in the calculator and then use the exponent key to solve for the variable.

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

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

  • Modeling population growth
  • Modeling chemical reactions
  • Modeling financial growth
  • Modeling electrical circuits

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

A: To use exponential equations to model real-world problems, you need to identify the variables and the relationships between them. For example, to model population growth, you would use an exponential equation to model the growth of the population over time.

Q: What are some common types of exponential equations?

A: Some common types of exponential equations include:

  • Linear exponential equations
  • Quadratic exponential equations
  • Polynomial exponential equations
  • Rational exponential equations

Q: How do I solve linear exponential equations?

A: To solve linear exponential equations, you need to rewrite the equation in terms of the same base and then set the exponents equal to each other. For example, to solve the equation $2^3x = 8$, you would rewrite it as $(23)x = 2^3$ and then set the exponents equal to each other.

Q: How do I solve quadratic exponential equations?

A: To solve quadratic exponential equations, you need to rewrite the equation in terms of the same base and then set the exponents equal to each other. For example, to solve the equation $2{3x2} = 8$, you would rewrite it as $(23){x^2} = 2^3$ and then set the exponents equal to each other.

Q: How do I solve polynomial exponential equations?

A: To solve polynomial exponential equations, you need to rewrite the equation in terms of the same base and then set the exponents equal to each other. For example, to solve the equation $2{3x3} = 8$, you would rewrite it as $(23){x^3} = 2^3$ and then set the exponents equal to each other.

Q: How do I solve rational exponential equations?

A: To solve rational exponential equations,