Rationalizing the Denominator Put each fractional expression...

Rationalizing the Denominator Put each fractional expression into standard form by rationalizing the denominator.
$$ (a)
\frac{1}{\sqrt{6}} \quad \text { (b) } \sqrt{\frac{3}{2}} \quad \text { (c) } \frac{9}{\sqrt[4]{2}}
$$

Key Concepts

Properties of Exponents and Rational Exponents

Radical expressions can be rewritten in terms of rational exponents, which can facilitate the simplification process. The laws of exponents allow for operations such as multiplication, division, and raising to a power to be applied more straightforwardly, which is particularly helpful when converting between radical and exponential forms.

Standard Form for Radical Expressions

Standard form refers to the convention of writing expressions in a fully simplified manner, typically with no radicals in the denominator. Converting an expression into standard form often requires rationalizing the denominator, ensuring that the expression is both simplified and presented in a standardized, easily recognizable format.

Rationalizing the Denominator

This concept involves eliminating any radicals or irrational numbers from the denominator of a fraction. The process typically includes multiplying the numerator and denominator by a suitable form of 1 (often involving the radical in question) so that the denominator becomes a rational number, thereby making the expression easier to interpret and compare.

Radical Expressions

Radical expressions include terms with roots, such as square roots, cube roots, or higher order roots. Understanding the properties of these expressions is essential for simplifying them correctly. This includes knowing how to multiply, divide, and combine radical expressions, as well as converting between radical notation and rational exponents when necessary.

Transcript

00:01 So for this question, we are given three radical expressions and we're asked to rationalise them.

00:07 Okay, so the way we're going to start off in part a, we've got 1 over root 6.

00:11 The way we're going to rationalise this is we're going to times this by a fraction that's going to be equal to 1, but that will allow us to get the root off of the bottom.

00:20 So that fraction here, it's going to be root 6 over root 6, because you'll see here that when we times this through, this root 6 times root 6 on the bottom, that's you're going to turn to 6.

00:31 So we've got the fractions off the bottom.

00:33 Now we just need to simplify the top.

00:35 And on the top, we got 1 times root 6.

00:36 That's going to be root 6 on the top.

00:39 There we go.

00:40 So for part 2 here, we've got all rooted 3 over 2.

00:46 What i'm going to do here is i'm going to bring the root into the fraction.

00:50 So it's going to be root 3 on the top and root 2 on the bottom.

00:55 Then we need to times by a fraction again that's going to be equal to 1, but will allow us to get the root off of the bottom here.

01:01 So you've got root 2 on the bottom here.

01:03 So in order to get root 2 off of the bottom, we're going to times that by root 2.

01:07 And in order to make the fraction 1, we're going to have root 2 on the top as well.