Solving Square Root Expressions: A Step-by-Step Guide

In this guide, you will learn a detailed method for simplifying complex square root expressions involving nested radicals. Specifically, we will solve the expression sqrt{11 - 6 sqrt{2}} by expressing it in the form sqrt{a} - sqrt{b}.

Introduction

Understanding and simplifying expressions involving square roots is a crucial skill in algebra and calculus. This guide aims to provide a clear and comprehensive approach to solving such expressions step by step.

The Problem

Consider the expression sqrt{11 - 6 sqrt{2}}. Our goal is to simplify this expression by expressing it in the form sqrt{a} - sqrt{b}.

Step 1: Expressing the Equation in a Standard Form

Let us assume that:

sqrt{11 - 6 sqrt{2}} sqrt{a} - sqrt{b}

Squaring both sides of the equation, we get:

11 - 6 sqrt{2} a b - 2sqrt{ab}

Step 2: Equating Rational and Irrational Parts

To proceed, we need to separate the rational and irrational parts of the equation:

Rational Part: Equate the rational parts of both sides:

a b 11

Irrational Part: Equate the irrational parts of both sides:

-2sqrt{ab} -6sqrt{2} sqrt{ab} 3sqrt{2} ab 18

Step 3: Solving the System of Equations

Now we have a system of equations:

a b 11

ab 18

We can treat a and b as the roots of the quadratic equation:

x^2 - (a b)x ab 0 x^2 - 11x 18 0

Solving this quadratic equation using the quadratic formula:

x (-b ± sqrt{b^2 - 4ac}) / 2a (11 ± sqrt{11^2 - 4 cdot 1 cdot 18}) / 2

This simplifies to:

x (11 ± sqrt{121 - 72}) / 2 (11 ± sqrt{49}) / 2 (11 ± 7) / 2

Thus, we have:

x (18 / 2) 9 x (4 / 2) 2

So, we can write:

a 9 b 2 or vice versa.

Step 4: Substituting Back

Substituting these values back, we get:

sqrt{11 - 6 sqrt{2}} sqrt{9} - sqrt{2} 3 - sqrt{2}

Final Answer

Therefore, the solution is:

sqrt{11 - 6 sqrt{2}} 3 - sqrt{2}

Intuitive Approach

Another intuitive way to solve this problem is to recognize a familiar pattern:

11 - 6 sqrt{2} 9 2 - 2*3*sqrt{2} (3 - sqrt{2})^2

Thus:

sqrt{11 - 6 sqrt{2}} 3 - sqrt{2}

Conclusion

Through the systematic approach detailed in this guide, we can effectively simplify complex square root expressions by expressing them in the form of sqrt{a} - sqrt{b}. This method not only simplifies the expression but also provides a clear understanding of the underlying algebraic principles.