Question

Why is root 2 over 2 equal to one over root 2?

Answer 1

The equality of
`\(√(2)/2\) and \(1/√(2)\)` is shown by rationalizing the denominator of the latter, which involves multiplying the numerator and denominator by
`\(√(2)\)` thereby transforming it into the former while maintaining its value.

Step-by-step explanation:

The question at hand is about simplifying radical expressions, specifically why
`\(√(2)/2\)` is equal to
`\(1/√(2)\)`. To understand this, we need to look at the properties of radicals and fractions. When we simplify a fraction, we aim to find an equivalent fraction with a simpler form. In this case, the trick lies in rationalizing the denominator. To rationalize the denominator of
`\(1/√(2)\)` we could multiply both the numerator and the denominator by
`\(√(2)\)`. This process of multiplying by the same quantity on the top and the bottom ensures that the overall value does not change (since multiplying by
`1` does not alter the value of a number) but it changes the appearance of the fraction. When we multiply
`\(1/√(2)\)`by
`\(√(2)/√(2)\`, we end up with
`\(√(2)/2\)`, which is the same as the original expression.

Thus, multiplying by a cleverly disguised form of 1 (in this case,

`\(√(2)/√(2)\))`, preserves equality and rationalizes the denominator, showing how the two expressions
`\(√(2)/2\)` and
`\(1/√(2)\)` are indeed equal.