Eliminating the Fourth Root

Oct 13, 2025 by miodeya.com 123 views

Introduction

In this article, we will be solving for the value of n in the function h(n) = 3, where h(n) is defined as the fourth root of 3n - 11. This problem involves algebraic manipulation and solving a quadratic equation. We will start by setting up the equation and then proceed to solve for n.

Setting Up the Equation

The function h(n) is defined as:

h(n) = √[4](3n - 11)

We are given that h(n) = 3, so we can set up the equation:

√[4](3n - 11) = 3

Eliminating the Fourth Root

To eliminate the fourth root, we can raise both sides of the equation to the power of 4:

(√[4](3n - 11))^4 = 3^4

This simplifies to:

3n - 11 = 81

Solving for n

Now we can solve for n by adding 11 to both sides of the equation:

3n = 81 + 11

3n = 92

Next, we can divide both sides of the equation by 3:

n = 92/3

Simplifying the Fraction

The fraction 92/3 can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 1. However, we can simplify it further by dividing both the numerator and the denominator by 1 and then reducing the fraction to its simplest form.

n = 92/3

Conclusion

In this article, we solved for the value of n in the function h(n) = 3, where h(n) is defined as the fourth root of 3n - 11. We started by setting up the equation and then proceeded to eliminate the fourth root by raising both sides of the equation to the power of 4. We then solved for n by adding 11 to both sides of the equation and dividing both sides by 3. The final answer is n = 92/3.

Final Answer

n = 92/3

Introduction

In our previous article, we solved for the value of n in the function h(n) = 3, where h(n) is defined as the fourth root of 3n - 11. In this article, we will provide a Q&A section to help clarify any questions or doubts that readers may have.

Q: What is the definition of the function h(n)?

A: The function h(n) is defined as the fourth root of 3n - 11. This means that h(n) = √[4](3n - 11).

Q: How do we eliminate the fourth root in the equation?

A: To eliminate the fourth root, we can raise both sides of the equation to the power of 4. This is because the fourth root and the fourth power are inverse operations.

Q: Why do we add 11 to both sides of the equation?

A: We add 11 to both sides of the equation to isolate the term 3n. This allows us to solve for n by dividing both sides of the equation by 3.

Q: Can we simplify the fraction 92/3?

A: Yes, the fraction 92/3 can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 1. However, in this case, the fraction is already in its simplest form.

Q: What is the final answer for n?

A: The final answer for n is 92/3.

Q: Why is it important to provide an exact answer in fraction form?

A: Providing an exact answer in fraction form is important because it ensures that the answer is precise and accurate. Rounding the answer can lead to errors and inaccuracies.

Q: Can we use a calculator to solve for n?

A: While a calculator can be used to solve for n, it is not necessary in this case. The solution can be found using algebraic manipulation and basic arithmetic operations.

Q: What is the significance of the function h(n) = 3?

A: The function h(n) = 3 is significant because it provides a specific value for the function h(n) in terms of n. This allows us to solve for n and understand the relationship between the function and the variable n.

Q: Can we use the function h(n) = 3 to solve for other values of n?

A: Yes, the function h(n) = 3 can be used to solve for other values of n by substituting different values of n into the equation and solving for the corresponding value of h(n).

Q: What is the next step after solving for n?

A: After solving for n, the next step is to use the value of n to solve for other related values or to apply the solution to a real-world problem.

Q: Can we use the solution to solve for other related problems?

A: Yes, the solution can be used to solve for other related problems by applying the same algebraic manipulation and basic arithmetic operations.

Q: What is the importance of solving for n in the function h(n) = 3?

A: Solving for n in the function h(n) = 3 is important because it provides a specific value for the function h(n) in terms of n, which can be used to solve for other related values or to apply the solution to a real-world problem.

Q: Can we use the solution to solve for other functions?

A: Yes, the solution can be used to solve for other functions by applying the same algebraic manipulation and basic arithmetic operations.

Q: What is the final answer for n in the function h(n) = 3?

A: The final answer for n in the function h(n) = 3 is 92/3.