Expressing Both Sides of the Equation in Terms of the Same Base

Oct 13, 2025 by windailys.com 12 views

Introduction

When solving equations involving exponents, it is often helpful to express both sides of the equation in terms of the same base. This allows us to use the properties of exponents to simplify the equation and solve for the variable. In this article, we will show how to solve the equation $$49^{3x} = 343^{2x+1}$$ by writing each side of the equation in terms of the same base.

Expressing Both Sides of the Equation in Terms of the Same Base

The first step in solving the equation is to express both sides of the equation in terms of the same base. We can do this by recognizing that both 49 and 343 are powers of 7. Specifically, we have:

49 = 7^2

343 = 7^3

Using these expressions, we can rewrite the equation as:

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

Simplifying the Equation

Now that we have expressed both sides of the equation in terms of the same base, we can use the properties of exponents to simplify the equation. Specifically, we can use the property that states:

(a^m)^n = a^{mn}

Using this property, we can simplify the equation as follows:

7^{2 \cdot 3x} = 7^{3 \cdot (2x+1)}

Equating the Exponents

Since the bases of the two expressions are the same, we can equate the exponents. This gives us the equation:

2 \cdot 3x = 3 \cdot (2x+1)

Solving for x

Now that we have equated the exponents, we can solve for x. To do this, we can start by distributing the 3 on the right-hand side of the equation:

6x = 6x + 3

Subtracting 6x from Both Sides

Next, we can subtract 6x from both sides of the equation to get:

0 = 3

Conclusion

Unfortunately, the equation $$0 = 3$$ is a contradiction, which means that there is no solution to the original equation. This is because the left-hand side of the equation is always 0, while the right-hand side of the equation is always 3. Therefore, we can conclude that the original equation has no solution.

Alternative Solution

However, we can also try to solve the equation by using a different approach. Specifically, we can try to solve the equation by using the property that states:

a^m = a^n \implies m = n

Using this property, we can rewrite the equation as:

49^{3x} = 343^{2x+1} \implies 7^{6x} = 7^{6x+3}

Equating the Exponents

Since the bases of the two expressions are the same, we can equate the exponents. This gives us the equation:

6x = 6x + 3

Subtracting 6x from Both Sides

Next, we can subtract 6x from both sides of the equation to get:

0 = 3

Conclusion

Final Conclusion

In conclusion, we have shown that the equation $$49^{3x} = 343^{2x+1}$$ has no solution. This is because the equation can be rewritten as $$7^{6x} = 7^{6x+3}$$, which is a contradiction. Therefore, we can conclude that the original equation has no solution.

Frequently Asked Questions

Q: What is the solution to the equation $$49^3x} = 343^{2x+1}$$? A The equation $$49^{3x = 343^{2x+1}$$ has no solution.
Q: Why is the equation $$49^3x} = 343^{2x+1}$$ a contradiction? A The equation $$49^{3x = 343^{2x+1}$$ is a contradiction because it can be rewritten as $$7^{6x} = 7^{6x+3}$$, which is a contradiction.
Q: What is the property that states $$a^m = a^n \implies m = n$$? A: The property that states $$a^m = a^n \implies m = n$$ is a fundamental property of exponents that allows us to equate the exponents of two expressions with the same base.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by Michael Spivak
[3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Keywords

Exponents
Equations
Algebra
Calculus
Mathematics
Computer Science

Introduction

In our previous article, we showed how to solve the equation $$49^{3x} = 343^{2x+1}$$ by writing each side of the equation in terms of the same base. However, we also showed that the equation has no solution. In this article, we will answer some frequently asked questions about solving the equation $$49^{3x} = 343^{2x+1}$$.

Q&A

Q: What is the solution to the equation $$49^{3x} = 343^{2x+1}$$?

A: The equation $$49^{3x} = 343^{2x+1}$$ has no solution.

Q: Why is the equation $$49^{3x} = 343^{2x+1}$$ a contradiction?

A: The equation $$49^{3x} = 343^{2x+1}$$ is a contradiction because it can be rewritten as $$7^{6x} = 7^{6x+3}$$, which is a contradiction.

Q: What is the property that states $$a^m = a^n \implies m = n$$?

A: The property that states $$a^m = a^n \implies m = n$$ is a fundamental property of exponents that allows us to equate the exponents of two expressions with the same base.

Q: Can we solve the equation $$49^{3x} = 343^{2x+1}$$ by using a different approach?

A: Yes, we can try to solve the equation $$49^{3x} = 343^{2x+1}$$ by using a different approach. However, we will still arrive at the same conclusion that the equation has no solution.

Q: What is the significance of the equation $$49^{3x} = 343^{2x+1}$$?

A: The equation $$49^{3x} = 343^{2x+1}$$ is a simple equation that can be used to illustrate the concept of exponents and how to solve equations involving exponents.

Q: Can we use the equation $$49^{3x} = 343^{2x+1}$$ to solve other equations?

A: Yes, we can use the concept of exponents and how to solve equations involving exponents to solve other equations.

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

A: Some common mistakes to avoid when solving equations involving exponents include:

Not using the correct property of exponents
Not simplifying the equation correctly
Not equating the exponents correctly

Q: How can we use the equation $$49^{3x} = 343^{2x+1}$$ to teach students about exponents?

A: We can use the equation $$49^{3x} = 343^{2x+1}$$ to teach students about exponents by:

Using the equation to illustrate the concept of exponents
Having students solve the equation and explain their reasoning
Using the equation to practice simplifying and equating exponents

Conclusion

In conclusion, we have answered some frequently asked questions about solving the equation $$49^{3x} = 343^{2x+1}$$. We have shown that the equation has no solution and have provided some common mistakes to avoid when solving equations involving exponents. We have also provided some ideas for using the equation to teach students about exponents.

Frequently Asked Questions

Q: What is the solution to the equation $$49^3x} = 343^{2x+1}$$? A The equation $$49^{3x = 343^{2x+1}$$ has no solution.
Q: Why is the equation $$49^3x} = 343^{2x+1}$$ a contradiction? A The equation $$49^{3x = 343^{2x+1}$$ is a contradiction because it can be rewritten as $$7^{6x} = 7^{6x+3}$$, which is a contradiction.
Q: What is the property that states $$a^m = a^n \implies m = n$$? A: The property that states $$a^m = a^n \implies m = n$$ is a fundamental property of exponents that allows us to equate the exponents of two expressions with the same base.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by Michael Spivak
[3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Keywords

Exponents
Equations
Algebra
Calculus
Mathematics
Computer Science