Addition and Subtraction of Square Roots

Adding and subtracting square roots is much like adding and subtracting fractions. To add or subtract fractions with a common denominator, we combine the numerators and keep the common denominator. For example,

[latex]\frac{2}{\color{blue}{7}}+\frac{3}{\color{blue}{7}}=\frac{2+3}{\color{blue}{7}}=\frac{5}{\color{blue}{7}}[/latex].

In words, this would be:

[latex]2[/latex] sevenths + [latex]3[/latex] sevenths = [latex](2+3)[/latex] sevenths = [latex]5[/latex] sevenths.

Suppose instead of sevenths, we have oranges:

[latex]4[/latex] oranges + [latex]3[/latex] oranges = [latex](4 + 3)[/latex] oranges = [latex]7[/latex] oranges.

We can add and subtract expressions with a common item such as these by combining the numbers and keeping the common item. Terms that have a common item like this are called like terms.

We do exactly the same thing to add or subtract common square roots: we add or subtract the numbers in front of the radicals and keep the common radical.

[latex]5\color{blue}{\sqrt{3}}+7\color{blue}{\sqrt{3}}=(5+7)\color{blue}{\sqrt{3}}=12\color{blue}{\sqrt{3}}[/latex]

The key to combining square roots by addition or subtraction is look at the radicand. If these are the same, then addition and subtraction are possible. If not, then we cannot combine the two radicals.

It may help to think of square root terms with words when you are adding and subtracting them. The last example could be read “[latex]3[/latex] square roots of eleven plus [latex]7[/latex] square roots of eleven equals [latex]10[/latex] square roots of eleven”.

This next example contains more addends, or terms that are being added together. Notice how we can combine like terms (square roots that have the same root) but we cannot combine unlike terms.

Notice that the expression in the previous example is simplified even though it has two terms: [latex]3\sqrt{2}[/latex] and [latex]5\sqrt{3}[/latex]. It would be a mistake to try to combine them further! [latex]\sqrt{2}[/latex] and [latex]\sqrt{3}[/latex] are not like radicands so they cannot be added. Notice also that [latex]\sqrt{3}=1\sqrt{3}[/latex].

 

 

 

Sometimes it appears that the radicals cannot be combined but after we simplify them it turns out that they can be combined.

 

The video, shows more examples of adding and subtracting square roots.

Multiplying and Dividing Square Roots

Previously, we have encountered both the product rule and the quotient rule for square roots. These rules can also be applied backwards.

We can use these rules in reverse to multiply and divide square roots.

Notice that both [latex]\sqrt{18}[/latex] and [latex]\sqrt{16}[/latex] can be written as products involving perfect square factors. In the next example, we will use the same product from above to show that we can simplify before multiplying and get the same result.

In both examples, we arrive at the same product, [latex]12\sqrt{2}[/latex]. It does not matter whether we multiply the radicands then simplify or simplify each radical first them multiply.

 

 

Notice that the fraction [latex]\frac{640}{40}[/latex] can be simplified before taking the square root. The next example explores simplifying the fraction first.

As with multiplication, the main idea here is that sometimes it makes sense to divide and then simplify, and other times it makes sense to simplify and then divide. Whichever order we choose, though, we should arrive at the same answer.

Rationalizing Denominators

In cases where we have a fraction with a radical in the denominator, we can use a technique called rationalizing the denominator to eliminate the radical from the denominator. As we will see, the radical does not simply vanish, but rather it moves from the denominator to the numerator of the fraction. Historically, the point of rationalizing a denominator was to make it possible to divide. This was long before calculators and computers existed, when people routinely performed division by hand. Dividing by a rational number was common, whereas dividing by an irrational number was impossible. Today it has become a standard form for having an exact irrational number with a radical.

The idea of rationalizing a denominator makes a bit more sense if you consider the definition of “rationalize”. The numbers [latex]5[/latex], [latex]\frac{1}{2}[/latex], and [latex]0.75[/latex] are all rational numbers—they can each be expressed as a ratio of two integers [latex]\frac{5}{1},\frac{1}{2}[/latex], and [latex]\frac{3}{4}[/latex] respectively. Some radicals are irrational numbers because they cannot be represented as a ratio of two integers. As a result, the point of rationalizing a denominator is to change the expression so that the denominator becomes a rational number.

Rationalizing Denominators with One Term

Consider the fraction [latex]\frac{1}{\sqrt{2}}[/latex]. Its denominator is [latex]\sqrt{2}[/latex], an irrational number.

We can build an equivalent fraction without changing its value if we multiply it by a quantity equal to [latex]1[/latex]. In this case, let’s multiply by [latex]\frac{\sqrt{2}}{\sqrt{2}}[/latex]. We multiply the numerator and denominator by [latex]\sqrt{2}[/latex]. Watch what happens.

[latex]\frac{1}{\sqrt{2}}\cdot \color{blue}{1}=\frac{1}{\sqrt{2}}\cdot \color{blue}{\frac{\sqrt{2}}{\sqrt{2}}}=\frac{1\cdot\sqrt{2}}{\sqrt{2}\cdot\sqrt{2}}=\frac{\sqrt{2}}{\sqrt{2\cdot 2}}=\frac{\sqrt{2}}{\sqrt{4}}=\frac{\sqrt{2}}{2}[/latex]

The denominator of the new fraction is no longer a radical. Notice, however, that the numerator is!

So why choose to multiply [latex]\frac{1}{\sqrt{2}}[/latex] by [latex]\frac{\sqrt{2}}{\sqrt{2}}[/latex]? We know that the square root of a perfect square is a whole number. So we want to turn the number under the radical on the denominator into a perfect square, [latex]4[/latex].  To turn [latex]2[/latex] into [latex]4[/latex] we must multiply by [latex]2[/latex]. But the [latex]2[/latex] on the denominator is under a square root, so we must multiply by [latex]\sqrt{2}[/latex]. And, if we multiply the denominator by [latex]\sqrt{2}[/latex] we must also multiply the numerator by [latex]\sqrt{2}[/latex] to make an equivalent fraction.

Here are some more examples. Notice how the value of the fraction is not changed at all—it is simply being multiplied by [latex]1[/latex].