Question

question 2
part a
question
a rectangular prism has side lengths of $2\sqrt{6}$ cm, $\sqrt{2}$ cm, and $2\sqrt{3}$ cm.

without using a calculator, put these side lengths in order from greatest to least. (hint: consider the value of each radicand and what taking the square root does to each value.)
drag each tile to the correct box.
tiles
$2\sqrt{6}$ cm $\sqrt{2}$ cm $2\sqrt{3}$ cm

Explanation:

Step1: Simplify each term

To compare the side lengths, we can square each term (since squaring is a monotonically increasing function for non - negative numbers, so if \(a>b\geq0\), then \(a^{2}>b^{2}\))

For \(2\sqrt{6}\): \((2\sqrt{6})^{2}=2^{2}\times(\sqrt{6})^{2}=4\times6 = 24\)

For \(\sqrt{2}\): \((\sqrt{2})^{2}=2\)

For \(2\sqrt{3}\): \((2\sqrt{3})^{2}=2^{2}\times(\sqrt{3})^{2}=4\times3=12\)

Step2: Compare the squared values

We have the squared values: \(24\) (for \(2\sqrt{6}\)), \(2\) (for \(\sqrt{2}\)) and \(12\) (for \(2\sqrt{3}\))

Since \(24>12 > 2\), and because we squared non - negative numbers, the original numbers satisfy \(2\sqrt{6}>2\sqrt{3}>\sqrt{2}\)

Answer:

From greatest to least: \(2\sqrt{6}\text{ cm}\), \(2\sqrt{3}\text{ cm}\), \(\sqrt{2}\text{ cm}\)