When we are trying to find the square root of a number (say, [latex]25[/latex]), we are trying to find a number that when multiplied by itself gives that original number. In the case of [latex]25[/latex], we find that [latex]5\cdot5=25[/latex], so [latex]5[/latex] is the square root of [latex]25[/latex].

The square root is the inverse of the square (exponent of 2), much like multiplication is the inverse of division. A perfect square is any whole number that has been squared. Consequently, all perfect squares have square roots that are whole numbers.

The symbol for square root is the radical sign [latex]\sqrt{\hphantom{5}}[/latex]. So [latex]\sqrt{25}=5[/latex]. The number under the radical sign is called the radicand. Because we are “unsquaring” a number when we find a square root, the radicand must be positive. This is because there is no real number that multiplies by itself to result in a negative number.

Consider [latex] \sqrt{25}[/latex] again. If we think about it, there is another number that, when multiplied by itself, also results in [latex]25[/latex]. That number is [latex]−5[/latex].

[latex] \begin{array}{r}5\cdot 5=25\\-5\cdot -5=25\end{array}[/latex]

By definition, the square root symbol, [latex]\sqrt{\hphantom{5}}[/latex], always means to find the positive root called the principal root. So while [latex]5\cdot5[/latex] and [latex]−5\cdot−5[/latex] both equal [latex]25[/latex], only [latex]5[/latex] is the principal root. If we want to find the negative square root, we must place a negative sign infant of the radical: [latex]-\sqrt{25}=-5[/latex]. Zero is special because it has only one square root:  [latex]\sqrt{0}=0[/latex]).

The notation that we use to express a square root for any real number, [latex]a[/latex], is as follows:

The last problem in the previous example shows us an important relationship between squares and square roots.

Remember that  [latex]\big |\,a\,\big |[/latex] represents the distance of the number [latex]a[/latex] from zero.  [latex]\big |\,a\,\big | \ge 0[/latex]. Consequently, [latex]\sqrt{6^2}=\big |\,6\,\big |=6[/latex] and [latex]\sqrt{(-6)^2}=\big |\,-6\,\big |=6[/latex].

In the video that follows, we simplify more square roots using the fact that [latex]\sqrt{a^2}=\big |\,a\,\big |[/latex] means finding the principal square root.

Finding Square Roots Using Factoring

We have previously considered the square root of a fraction. For example, [latex]\sqrt\frac{4}{9}=\frac{2}{3}[/latex] because [latex]\frac{2}{3}\cdot\frac{2}{3}=\frac{4}{9}[/latex]. We also discovered the Quotient Rule for Square Roots and were able to simply the square root of a fraction by taking the square root of the numerator and denominator separately. For example,  [latex]\sqrt\frac{4}{9}=\frac{\sqrt{4}}{\sqrt{9}}=\frac{2}{3}[/latex].

A similar rule applies to products. After all, division is just multiplication by the reciprocal, so the quotient rule could be considered a product rule of the reciprocal: [latex]\sqrt{\frac{a}{b}}=\sqrt{a\cdot\frac{1}{b}}[/latex] and [latex]\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{a}\cdot\frac{1}{\sqrt{b}}[/latex].

This is useful if we are working with a number whose square we do not know right away.  We can use factoring and the product rule for square roots to find square roots such as [latex]\sqrt{144}[/latex], or  [latex]\sqrt{225}[/latex].

Let’s look at some more examples of expressions with square roots.  Pay particular attention to how number 3 is evaluated.

The last example we showed reminds us of an important characteristic of square roots. We can only take the square root of values that are non-negative.

Simplifying Square Roots

So far, we have seen examples that are perfect squares. That is, each is a number whose square root is a whole number. But many radical expressions are not perfect squares. Some of these radicals can still be simplified by finding perfect square factors. The example below illustrates how to factor the radicand, looking for pairs of factors that can be expressed as a square.

The final answer [latex] 3\sqrt{7}[/latex] may look a bit odd, but it is in simplified form. We read this as “three radical seven” or “three times the square root of seven.”

In the next example, we make use of the common squares we know, instead of using prime factors. It helps to have the squares of the numbers between [latex]0[/latex] and [latex]10[/latex] fresh in your mind to make simplifying radicals faster.

• [latex]0^2=0[/latex]
• [latex]2^2=4[/latex]
• [latex]3^2=9[/latex]
• [latex]4^2=16[/latex]
• [latex]5^2=25[/latex]
• [latex]6^2=36[/latex]
• [latex]7^2=49[/latex]
• [latex]8^2=64[/latex]
• [latex]9^2=81[/latex]
• [latex]10^2=100[/latex]

In this last video, we show examples of simplifying radicals that are not perfect squares.

Decimal Approximations

All of the radicals we have simplified have resulted in whole number answers or have natural numbers left under the radical after simplifying. These are all exact values of square roots. However, it may not be necessary or convenient to have an exact answer. If we were to ask a carpenter to cut a board [latex]7\sqrt{3}[/latex] inches long, they would tell us to take a hike. Consequently, to get an approximation we can use the radical button on a calculator.

How we enter a square root on a calculator depends on the calculator. But it is usually the [latex]\sqrt{\hphantom{5}}[/latex] button followed by [latex]3[/latex] or [latex]3[/latex] followed by the [latex]\sqrt{\hphantom{5}}[/latex] button.

In the free online Desmos calculator shown here, it is the [latex]\sqrt{\hphantom{5}}[/latex] button followed by [latex]3[/latex]. We can also calculate [latex]7\sqrt{3}[/latex] by using the 7 button, the times button, the square root button and the three button. The results are shown as [latex]\sqrt{3}=1.732050808[/latex] and [latex]7\sqrt{3}=12.12435565[/latex]. Both of these are approximations. The exact values are are irrational numbers: decimals that never end and never repeat. We can round these decimals to whatever place value is asked for or makes sense. In the case of the board that needs to be cut to [latex]7\sqrt{3}[/latex] inches, rounding to the tenths place is sufficient: [latex]7\sqrt{3}=12.1[/latex].

Rounding a decimal to a certain place value requires us to identify where in the decimal number that place value lies, then looking at the digit immediately after it to decide whether to leave the place value digit alone or to increase it by one.

example

Round [latex]18.379[/latex] to the nearest

1. 1. tenth
   2. whole number

Show Solution

Solution

1. Round [latex]18.379[/latex] to the nearest tenth.
[latex]18.379[/latex]
Locate the tenths place and mark it with an arrow.
Underline the digit to the right of the tenths digit.
Because [latex]7[/latex] is greater than or equal to [latex]5[/latex], add [latex]1[/latex] to the [latex]3[/latex].
Rewrite the number, deleting all digits to the right of the tenths place.	[latex]18.4[/latex]
So, [latex]18.379[/latex] rounded to the nearest tenth is [latex]18.4[/latex].

2. Round [latex]18.379[/latex] to the nearest whole number.
[latex]18.379[/latex]
Locate the ones place and mark it with an arrow.
Underline the digit to the right of the ones place.
Since [latex]3[/latex] is not greater than or equal to [latex]5[/latex], do not add [latex]1[/latex] to the [latex]8[/latex].
Rewrite the number, deleting all digits to the right of the ones place.	[latex]18[/latex]
So [latex]18.379[/latex] rounded to the nearest whole number is [latex]18[/latex].

Watch the following video to see an example of how to round a number to several different place values.

Summary

The square root of a number is the number which, when multiplied by itself, gives the original number. Principal square roots are always positive and the square root of [latex]0[/latex] is [latex]0[/latex]. You can only take the square root of values that are nonnegative. The square root of a perfect square will be a whole number. Other square roots can be simplified by identifying factors that are perfect squares and taking their square root.  Square roots that are not perfect can also be estimated by using a calculator and rounding to an appropriate place value.