5.7 Simplify and use square roots

Square root of a negative number

Can we simplify $\sqrt{\mathrm{-25}}?$ Is there a number whose square is $\mathrm{-25}?$

None of the numbers that we have dealt with so far have a square that is $\mathrm{-25}.$ Why? Any positive number squared is positive, and any negative number squared is also positive. In the next chapter we will see that all the numbers we work with are called the real numbers. So we say there is no real number equal to $\sqrt{\mathrm{-25}}.$ If we are asked to find the square root of any negative number, we say that the solution is not a real number.

Square roots and the order of operations

When using the order of operations to simplify an expression that has square roots, we treat the radical sign as a grouping symbol. We simplify any expressions under the radical sign before performing other operations.

Notice the different answers in parts ⓐ and ⓑ of [link] . It is important to follow the order of operations correctly. In ⓐ , we took each square root first and then added them. In ⓑ , we added under the radical sign first and then found the square root.

Estimate square roots

So far we have only worked with square roots of perfect squares. The square roots of other numbers are not whole numbers.

We might conclude that the square roots of numbers between $4$ and $9$ will be between $2$ and $3,$ and they will not be whole numbers. Based on the pattern in the table above, we could say that $\sqrt{5}$ is between $2$ and $3.$ Using inequality symbols, we write

Estimate $\sqrt{60}$ between two consecutive whole numbers.

Solution

Think of the perfect squares closest to $60.$ Make a small table of these perfect squares and their squares roots.

$\text{Locate 60 between two consecutive perfect squares.}$	$49<60<64$
$\sqrt{60}\text{is between their square roots.}$	$7<\sqrt{60}<8$

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Approximate square roots with a calculator

There are mathematical methods to approximate square roots, but it is much more convenient to use a calculator to find square roots. Find the $\sqrt{\phantom{0}}$ or $\sqrt{x}$ key on your calculator. You will to use this key to approximate square roots. When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact number. It is an approximation, to the number of digits shown on your calculator’s display. The symbol for an approximation is $\approx$ and it is read approximately .