Rationalize the denominator.
$\(\frac{1}{\sqrt7}\)$

Verified step by step guidance

1. Identify the expression to rationalize: \(\frac{1}{\sqrt7}\), where the denominator contains a square root.
2. To rationalize the denominator, multiply both the numerator and the denominator by \(\sqrt7\) to eliminate the square root in the denominator.
3. Write the multiplication as: \(\frac{1}{\sqrt7} \times \frac{\sqrt7}{\sqrt7}\).
4. Multiply the numerators together and the denominators together: numerator becomes \(1 \times \sqrt7 = \sqrt7\), denominator becomes \(\sqrt7 \times \sqrt7 = 7\).
5. Rewrite the expression with the rationalized denominator as \(\frac{\sqrt7}{7}\).

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any irrational numbers, such as square roots, from the denominator of a fraction. This is done by multiplying the numerator and denominator by a suitable expression that removes the radical, making the expression easier to interpret and use in further calculations.

Properties of Square Roots

Square roots represent a number which, when multiplied by itself, gives the original number. Understanding that √a × √a = a is essential for rationalizing denominators, as multiplying by the square root in the denominator can simplify the expression by removing the radical.

Multiplying Fractions by 1

Multiplying a fraction by a form of 1, such as √7/√7, does not change its value but can change its form. This technique is used to rationalize denominators by creating an equivalent fraction with a rational denominator, facilitating easier manipulation and simplification.