Question

The value of \(\sqrt{15-2 \sqrt{15-2 \sqrt{15-2 \sqrt{15 \ldots \infty}}}}\) (a) 6 (b) 5 (c) 4 (d) 3

Short answer

Answer: 5

Step-by-step solution

Define the expression with a variable

Let x be the value of the expression, so we have: x = \(\sqrt{15 - 2\sqrt{15 - 2\sqrt{15 - 2\sqrt{15 \ldots \infty}}}}\) Since the expression is an infinite nested radical, we can represent the inner radical as x as well.

Rewrite the expression with the variable

Now we can rewrite the expression as: x = \(\sqrt{15 - 2\sqrt{15 - 2\sqrt{15 - 2x}}}\)

Solve for the variable

To solve for x, we will do the following steps: 1. Square both sides of the equation to eliminate the square root: x^2 = \(15 - 2\sqrt{15 - 2\sqrt{15 - 2x}}\) 2. Isolate the inner radical by bringing the rest to the other side: \(2\sqrt{15 - 2\sqrt{15 - 2x}} = x^2 - 15\) 3. Square both sides again to eliminate the square root: \(4(15 - 2\sqrt{15 - 2x}) = (x^2-15)^2\) 4. Expanding the equation: \(60 - 8\sqrt{15 - 2x} = x^4 - 30x^2 + 225\) 5. Isolate the remaining radical: \(8\sqrt{15 - 2x} = x^4 - 30x^2 + 285 - 60\) 6. Rearrange the equation: \(8\sqrt{15 - 2x} = x^4 - 30x^2 + 225\) 7. Divide both sides by 8: \(\sqrt{15 - 2x} = \frac{x^4 - 30x^2 + 225}{8}\) 8. Square both sides one last time: \(15 - 2x = (\frac{x^4 - 30x^2 + 225}{8})^2\) 9. Simplify the equation: \(15 - 2x = \frac{x^8 - 60x^6 + 675x^4 - 18000x^2 + 50625}{64}\) 10. Multiply by 64 to clear the denominator: \(960 - 128x = x^8 - 60x^6 + 675x^4 - 18000x^2 + 50625\) 11. Transfer everything to one side to get a polynomial equation: \(x^8 - 60x^6 + 675x^4 - 18000x^2 + 50625 + 128x - 960 = 0\)

Check possible solutions

Now we can check possible solutions from the multiple-choice options. (a) If x = 6, the polynomial equation is not satisfied. (b) If x = 5, the polynomial equation is satisfied. (c) If x = 4, the polynomial equation is not satisfied. (d) If x = 3, the polynomial equation is not satisfied.

Conclusion

Since the polynomial equation is only satisfied when x = 5, we conclude that the value of the given expression is: \(\sqrt{15-2 \sqrt{15-2 \sqrt{15-2 \sqrt{15 \ldots \infty}}}}\) = 5 Therefore, the correct multiple choice option is (b) 5.

Key concepts

Polynomial Equation

When working with infinite nested radicals, we often encounter polynomial equations during the process of finding a solution. This occurs because we need to convert the complex radical structure into an equation that we can manipulate more easily. In the given exercise, we demonstrate how to define and handle such an equation.

To start, we define our variable, say \(x\), to represent the value of the entire expression. This allows us to express the repeated pattern within the radical using the same variable. By defining the problem this way, we translate the infinite process into a solvable format. Afterward, we use algebraic manipulations such as squaring and isolating terms to reformulate the expression into a polynomial equation format.

Eventually, the polynomial equation obtained in the exercise is:

• \(x^8 - 60x^6 + 675x^4 - 18000x^2 + 50625 + 128x - 960 = 0\)

This polynomial equation serves as a crucial step in narrowing down the possible values of \(x\) by testing the given multiple-choice options. Hence, the polynomial equation serves as a bridge, connecting the radical expression to a series of testable solutions.

Nested Radicals

Nested radicals involve layers of square roots within each other, which can extend infinitely or finitely. In our problem, we are dealing with an infinite nested radical. Understanding the concept of nested radicals is key to solving such exercises efficiently.

The given problem presents a pattern of repeated operations under the square root function, denoted as:

• \(\sqrt{15-2\sqrt{15-2\sqrt{15-2\sqrt{15 \ldots \infty}}}}\).

Handling nested radicals often involves recognizing the repetitive nature and simplifying it by defining it with a variable. This allows us to create a simplified, more manageable expression. From there, we make educated assumptions about the variable's value, typically by solving the resultant polynomial equation.

Nested radicals may appear daunting, but by breaking them down systematically, as shown in the solution, they become straightforward. In our example, iterating through subsequent steps transforms the complexity of the radical into a simpler polynomial form. This method is critical for unveiling the value concealed within these layers.

Mathematical Reasoning

Mathematical reasoning is the logical thought process essential for tackling problems, particularly those involving intricate mathematical structures. It enables us to navigate from the problem statement to the solution in a coherent way.

Here's how reasoning played a role in the given exercise:

• Defining the Variable: By deciding to equate the expression to \(x\), we employed reasoning to convert the infinite nested pattern into a form we can mathematically manipulate.
• Rewriting and Simplifying: The decision to square and rearrange the terms to simplify, showing cause and effect properly, relies on clear mathematical reasoning.
• Testing Options: With the polynomial equation in hand, reasoning allows us to validate which among the multiple-choice answers satisfies the equation fully.

Strong mathematical reasoning ensures each step logically follows the last, maintaining accuracy and clarity. It drives the method of exploring and solving mathematical curiosities and challenges, like nested radicals, with confidence.