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
$2 \sqrt{6} cm$
$\sqrt{2} cm$
$2 \sqrt{3} cm$
Sequence
$\square$
$\square > \square$
Answer (2)
Rewrite each side length in the form a by moving the coefficients inside the square root: 2 6 = 24 , 2 = 2 , and 2 3 = 12 .
Compare the values inside the square roots: 12 > 2"> 24 > 12 > 2 .
Order the original side lengths based on the comparison of the values inside the square roots: 2 \sqrt{3} > \sqrt{2}"> 2 6 > 2 3 > 2 .
State the final answer: 2 \sqrt{3} cm > \sqrt{2} cm}"> 2 6 c m > 2 3 c m > 2 c m .
Explanation
Analyze the problem and given data We are given three side lengths of a rectangular prism: $2
\sqrt{6} c m , \sqrt{2}$ cm, and 2 3 cm. Our goal is to order these lengths from greatest to least without using a calculator. To do this, we'll compare the values inside the square roots after manipulating the coefficients.
Rewrite side lengths with coefficients inside the square root First, let's rewrite each side length so that the coefficient is inside the square root. Recall that $a\sqrt{b} = \sqrt{a^2
b}$.
For 2 6 , we have $2 \sqrt{6} = \sqrt{2^2
\cdot 6} = \sqrt{4 \cdot 6} = \sqrt{24}$.
For 2 , the side length is already in the desired form.
For 2 3 , we have $2 \sqrt{3} = \sqrt{2^2
\cdot 3} = \sqrt{4 \cdot 3} = \sqrt{12}$.
So, we have the side lengths 24 cm, 2 cm, and 12 cm.
Compare the values inside the square roots and order the side lengths Now, we can easily compare the side lengths by comparing the values inside the square roots. Since 12 > 2"> 24 > 12 > 2 , we have \sqrt{12} > \sqrt{2}"> 24 > 12 > 2 .
Therefore, the order from greatest to least is 2 6 cm 2 \sqrt{3}"> > 2 3 cm \sqrt{2}"> > 2 cm.
State the final answer The side lengths in order from greatest to least are:
2 \sqrt{3} cm > \sqrt{2} cm"> 2 6 c m > 2 3 c m > 2 c m
Examples
Understanding how to compare and order radical expressions is useful in various real-world scenarios. For example, when constructing a building, engineers need to compare different lengths and distances, which may involve square roots. Suppose an architect is designing a rectangular garden with sides 2 6 meters and 2 3 meters. To optimize the layout, they need to know which side is longer. By converting these lengths to 24 meters and 12 meters, they can easily determine that the side with length 2 6 meters is longer, allowing them to plan the garden's dimensions effectively.
The side lengths in order from greatest to least are 2 \sqrt{3} \text{ cm} > \sqrt{2} \text{ cm}"> 2 6 cm > 2 3 cm > 2 cm . This is determined by comparing the values inside the square roots after moving the coefficients inside. The complete ordering is achieved by rewriting each side length appropriately and comparing them numerically.
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