Chapter A.1, Problem 75E

Solution

Replace | | with <, = , or > to make a true statement.

  $\sqrt{2+6}|<\sqrt{2}+\sqrt{6}$ .

Given:

The given statement is $\sqrt{2+6}\left|\right|\sqrt{2}+\sqrt{6}$ .

Concept Used:

• The greater than symbol means the number on the left is greater than the number on the right.
• The greater than or equal symbol means the number on the left is greater than or equal to the number on the right.
• The less than symbol means that the number on the left is less than the number on the right.

Calculation:

The given statement is $\sqrt{2+6}\left|\right|\sqrt{2}+\sqrt{6}$ .

  $\begin{array}{c}\sqrt{2+6}=\sqrt{8}\\ =\sqrt{2\cdot 2\cdot 2}\\ =2\sqrt{2}\\ =2(1.41)\\ =2.82\end{array}$

  $\begin{array}{c}\sqrt{2}+\sqrt{6}=1.41+2.45\\ =3.86\end{array}$

Therefore $\sqrt{2+6}|<\sqrt{2}+\sqrt{6}$

Thus $\sqrt{2+6}$ is less than $\sqrt{2}+\sqrt{6}$ .