Rationalize the denominator.
$\frac{\sqrt{2}}{\sqrt{5}}$

Verified step by step guidance

1. Identify the expression to rationalize: \(\frac{\sqrt{2}}{\sqrt{5}}\). The goal is to eliminate the square root from the denominator.
2. Multiply both the numerator and the denominator by \(\sqrt{5}\), which is the conjugate in this case, to rationalize the denominator. This gives: \(\frac{\sqrt{2}}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}\).
3. Use the property of square roots that \(\sqrt{a} \times \sqrt{a} = a\) to simplify the denominator: \(\sqrt{5} \times \sqrt{5} = 5\).
4. Multiply the numerators together: \(\sqrt{2} \times \sqrt{5} = \sqrt{2 \times 5} = \sqrt{10}\).
5. Write the new expression with the rationalized denominator: \(\frac{\sqrt{10}}{5}\). This is the simplified form with no radical in the denominator.

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any radicals (square roots) from the denominator of a fraction. This is done by multiplying the numerator and denominator by a suitable expression that will remove the radical from the denominator, making the expression easier to interpret and use.

Properties of Square Roots

Square roots have properties such as \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \) and \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \). Understanding these properties helps simplify expressions involving radicals and is essential when rationalizing denominators.

Multiplying by a Form of One

To rationalize a denominator, multiply the fraction by a form of one that contains the radical in the denominator, such as \( \frac{\sqrt{5}}{\sqrt{5}} \). This does not change the value of the expression but helps eliminate the radical from the denominator.