Solve For \[$ X \$\] In The Equation: \[$(2x + 1)^4 = 81\$\]

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Introduction

In this article, we will delve into solving for x in the equation (2x + 1)^4 = 81. This equation involves a fourth power, which can be challenging to solve. However, with the right approach and techniques, we can find the value of x that satisfies the equation. We will break down the solution step by step, making it easy to follow and understand.

Understanding the Equation

The given equation is (2x + 1)^4 = 81. To solve for x, we need to isolate x on one side of the equation. The first step is to take the fourth root of both sides of the equation. This will help us get rid of the exponent and simplify the equation.

Taking the Fourth Root

Taking the fourth root of both sides of the equation, we get:

(2x+1)44=814\sqrt[4]{(2x + 1)^4} = \sqrt[4]{81}

This simplifies to:

2x+1=8142x + 1 = \sqrt[4]{81}

Simplifying the Right-Hand Side

The right-hand side of the equation is 814\sqrt[4]{81}. To simplify this, we can rewrite 81 as a power of a prime number. Since 81 = 3^4, we can rewrite the equation as:

2x+1=32x + 1 = 3

Solving for x

Now that we have simplified the equation, we can solve for x. Subtracting 1 from both sides of the equation, we get:

2x=22x = 2

Dividing both sides of the equation by 2, we get:

x=1x = 1

Checking the Solution

To verify that x = 1 is the correct solution, we can substitute it back into the original equation. Plugging x = 1 into the equation (2x + 1)^4 = 81, we get:

(2(1)+1)4=(3)4=81(2(1) + 1)^4 = (3)^4 = 81

This confirms that x = 1 is indeed the correct solution.

Conclusion

In this article, we solved for x in the equation (2x + 1)^4 = 81. We took the fourth root of both sides of the equation, simplified the right-hand side, and solved for x. The final solution is x = 1. We also verified the solution by plugging it back into the original equation.

Additional Tips and Tricks

When solving equations involving exponents, it's essential to remember the following tips and tricks:

  • Take the root of both sides of the equation to get rid of the exponent.
  • Simplify the right-hand side of the equation by rewriting it in terms of prime numbers.
  • Solve for x by isolating x on one side of the equation.
  • Verify the solution by plugging it back into the original equation.

By following these tips and tricks, you can solve equations involving exponents with ease.

Common Mistakes to Avoid

When solving equations involving exponents, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Not taking the root of both sides of the equation.
  • Not simplifying the right-hand side of the equation.
  • Not solving for x by isolating x on one side of the equation.
  • Not verifying the solution by plugging it back into the original equation.

By avoiding these common mistakes, you can ensure that your solutions are accurate and reliable.

Real-World Applications

Solving equations involving exponents has many real-world applications. Here are a few examples:

  • Physics: Exponents are used to describe the motion of objects under the influence of gravity. For example, the equation for the height of an object under the influence of gravity is h = (1/2)gt^2, where h is the height, g is the acceleration due to gravity, and t is time.
  • Engineering: Exponents are used to describe the behavior of electrical circuits. For example, the equation for the voltage across a resistor is V = IR, where V is the voltage, I is the current, and R is the resistance.
  • Computer Science: Exponents are used to describe the time complexity of algorithms. For example, the equation for the time complexity of a sorting algorithm is O(n log n), where n is the number of elements.

By understanding how to solve equations involving exponents, you can apply this knowledge to a wide range of real-world problems.

Conclusion

Introduction

In our previous article, we solved for x in the equation (2x + 1)^4 = 81. We took the fourth root of both sides of the equation, simplified the right-hand side, and solved for x. The final solution is x = 1. In this article, we will answer some frequently asked questions about solving for x in the equation (2x + 1)^4 = 81.

Q: What is the first step in solving for x in the equation (2x + 1)^4 = 81?

A: The first step in solving for x in the equation (2x + 1)^4 = 81 is to take the fourth root of both sides of the equation. This will help us get rid of the exponent and simplify the equation.

Q: Why do we need to take the fourth root of both sides of the equation?

A: We need to take the fourth root of both sides of the equation because the exponent is 4. Taking the fourth root of both sides of the equation will help us get rid of the exponent and simplify the equation.

Q: How do we simplify the right-hand side of the equation?

A: We simplify the right-hand side of the equation by rewriting it in terms of prime numbers. In this case, we can rewrite 81 as 3^4.

Q: What is the next step in solving for x in the equation (2x + 1)^4 = 81?

A: The next step in solving for x in the equation (2x + 1)^4 = 81 is to solve for x by isolating x on one side of the equation. We can do this by subtracting 1 from both sides of the equation and then dividing both sides of the equation by 2.

Q: How do we verify the solution?

A: We verify the solution by plugging it back into the original equation. In this case, we plug x = 1 back into the equation (2x + 1)^4 = 81 and check if it is true.

Q: What are some common mistakes to avoid when solving for x in the equation (2x + 1)^4 = 81?

A: Some common mistakes to avoid when solving for x in the equation (2x + 1)^4 = 81 include:

  • Not taking the fourth root of both sides of the equation.
  • Not simplifying the right-hand side of the equation.
  • Not solving for x by isolating x on one side of the equation.
  • Not verifying the solution by plugging it back into the original equation.

Q: What are some real-world applications of solving for x in the equation (2x + 1)^4 = 81?

A: Some real-world applications of solving for x in the equation (2x + 1)^4 = 81 include:

  • Physics: Exponents are used to describe the motion of objects under the influence of gravity. For example, the equation for the height of an object under the influence of gravity is h = (1/2)gt^2, where h is the height, g is the acceleration due to gravity, and t is time.
  • Engineering: Exponents are used to describe the behavior of electrical circuits. For example, the equation for the voltage across a resistor is V = IR, where V is the voltage, I is the current, and R is the resistance.
  • Computer Science: Exponents are used to describe the time complexity of algorithms. For example, the equation for the time complexity of a sorting algorithm is O(n log n), where n is the number of elements.

Conclusion

In conclusion, solving for x in the equation (2x + 1)^4 = 81 involves taking the fourth root of both sides of the equation, simplifying the right-hand side, and solving for x. The final solution is x = 1. We also verified the solution by plugging it back into the original equation. By following the tips and tricks outlined in this article, you can solve equations involving exponents with ease.

Additional Resources

For more information on solving for x in the equation (2x + 1)^4 = 81, check out the following resources:

  • Mathway: A math problem solver that can help you solve equations involving exponents.
  • Khan Academy: A free online resource that provides video lessons and practice exercises on solving equations involving exponents.
  • Wolfram Alpha: A computational knowledge engine that can help you solve equations involving exponents.

By following these resources and tips, you can become proficient in solving equations involving exponents and apply this knowledge to a wide range of real-world problems.