Introduction to Finding √x and 1/√x for x3-2√2

In this article, we will explore the process of finding the value of the expression √x * 1/√x for a specific value of x3-2√2. This problem involves understanding and applying algebraic techniques to simplify and solve the expression. Let's break down the solution step-by-step.

Understanding the Expression

We need to find the value of √x * 1/√x. Let's substitute the given value of x3-2√2 into the expression.

Simplifying √x

First, we need to simplify the term √x for x3-2√2.

x 3 - 2√2 (√2 - 1)^2
(√2 - 1)(√2 - 1)
(2 - 2√2 1)
3 - 2√2

Taking the square root of both sides, we get:

√x √(3 - 2√2)
√((√2 - 1)^2)
√2 - 1

Simplifying 1/√x

Next, we simplify the term 1/√x where x3-2√2 and we already know that √x √2 - 1.

1/√x 1/(√2 - 1)

To avoid the complex conjugate, we multiply both the numerator and the denominator by the conjugate of the denominator:

1/(√2 - 1) * (√2 1)/(√2 1) (√2 1)/(√2^2 - 1^2)
(√2 1)/(2 - 1)
√2 1
(√2 - 1)

Combining the Terms

Now we combine the simplified forms of √x and 1/√x to find the final value of the expression:

√x * 1/√x (√2 - 1) * (√2 1)
(√2 - 1)^2
2 - 2√2 1
3 - 2√2
2√2
2√2

Conclusion

Thus, the value of √x * 1/√x for x3-2√2 is 2√2. This solution involves understanding and applying algebraic techniques such as simplifying nested square roots and working with algebraic expressions.

Key Points

To solve expressions involving nested square roots, follow these steps:

Identify the nested form and attempt to simplify it. Use algebraic identities to rewrite the terms. Multiply by the conjugate to simplify fractions involving square roots. Simplify and verify the result.