What is the simplest form of the radical expression sqrt 2 + sqrt 5/sqrt 2 - sqrt 5

Solution:

Given, the expression is sqrt 2 + sqrt 5 / sqrt 2 - sqrt 5

We have to find the simplest form of the radical expression.

The expression can be written as \(\frac{\sqrt{2}+\sqrt{5}}{\sqrt{2}-\sqrt{5}}\)

Multiplying both numerator and denominator with the conjugate of the denominator,

Conjugate of the denominator = \(\sqrt{2}+\sqrt{5}\)

The expression becomes \(\frac{\sqrt{2}+\sqrt{5}}{\sqrt{2}-\sqrt{5}}\times \frac{\sqrt{2}+\sqrt{5}}{\sqrt{2}+\sqrt{5}}\)

By using algebraic identity,

(a + b)(a + b) = (a + b)²

(a + b)² = a² + 2ab + b²

Numerator = \((\sqrt{2}+\sqrt{5})\times (\sqrt{2}+\sqrt{5})\)

= \((\sqrt{2}+\sqrt{5})^{2}\)

= \((\sqrt{2})^{2}+2(\sqrt{2}\sqrt{5})+(\sqrt{5})^{2}\)

= \(2+2\sqrt{10}+5\)

= 7 + 2√10

By using algebraic identity,

(a - b)(a + b) = (a² - b²)

Denominator = \((\sqrt{2}-\sqrt{5})(\sqrt{2}+\sqrt{5})\)

= \((\sqrt{2})^{2}-(\sqrt{5})^{2}\)

= 2 - 5

= -3

Now, \(\frac{\sqrt{2}+\sqrt{5}}{\sqrt{2}-\sqrt{5}}\) = \(\frac{7+2\sqrt{10}}{-3}\)

Therefore, the simplest form of the expression is \(\frac{7+2\sqrt{10}}{-3}\).

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What is the simplest form of the radical expression sqrt 2 + sqrt 5/sqrt 2 - sqrt 5

Summary:

The simplest form of the radical expression sqrt2 + sqrt5 / sqrt 2 - sqrt 5 is \(\frac{7+2\sqrt{10}}{-3}\).