Learning Outcomes
- Add and subtract square roots
- Multiply square roots
- Divide square roots
- Rationalize a single term denominator
Key words
- Radicand: the number under a radical
- Addends: terms that are added together
- Like Terms: terms that have a common item, like a square root with the same radicand
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.
Example
Simplify: [latex]3\sqrt{11}+7\sqrt{11}.[/latex]
The two square roots have the same radicand,[latex]\sqrt{11}.[/latex] This means we can combine them:
[latex]3\sqrt{11}+7\sqrt{11}[/latex]
[latex]=(3+7)\sqrt{11}[/latex]
[latex]=10\sqrt{11}[/latex]
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.
Example
Simplify: [latex]5\sqrt{2}+\sqrt{3}+4\sqrt{3}-2\sqrt{2}[/latex]
Rearrange terms so that like radicals are next to each other. Then add.
[latex]5\sqrt{2}-2\sqrt{2}+\sqrt{3}+4\sqrt{3}[/latex]
[latex]=(5-2)\sqrt{2}+(1+4)\sqrt{3}[/latex]
[latex]=3\sqrt{2}+5\sqrt{3}[/latex]
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].
Try It
Simplify: [latex]9\sqrt{5}-6\sqrt{7}-4\sqrt{5}+\sqrt{7}[/latex]
ADDING AND SUBTRACTING SQUARE ROOTS
For any real numbers [latex]a[/latex] and [latex]b[/latex], and whole number [latex]n[/latex], [latex]a\sqrt{n}+b\sqrt{n}=(a+b)\sqrt{n}[/latex] and [latex]a\sqrt{n}-b\sqrt{n}=(a-b)\sqrt{n}.[/latex]
To add or subtract square roots, the radicands must be identical.
Example
Subtract. [latex]5\sqrt{13}-3\sqrt{13}[/latex]
The radicands are the same, so these two radicals can be combined.
[latex]5\sqrt{13}-3\sqrt{13}[/latex]
[latex]=(5-3)\sqrt{13}[/latex]
[latex]=2\sqrt{13}[/latex]
Example
Simplify: [latex]4\sqrt{5}-\sqrt{3}-2\sqrt{5}[/latex]
Two of the square roots have the same radicand, so they can be combined.
[latex]4\sqrt{5}-\sqrt{3}-2\sqrt{5}[/latex]
[latex]=(4-2)\sqrt{5}-\sqrt{3}[/latex]
[latex]=2\sqrt{5}-\sqrt{3}[/latex]
The radicands are not the same—so they cannot be combined.
Try It
Simplify: [latex]-4\sqrt{7}-5\sqrt{11}+2\sqrt{7}[/latex]
Sometimes it appears that the radicals cannot be combined but after we simplify them it turns out that they can be combined.
Example
Simplify. [latex]2\sqrt{40}+\sqrt{250}[/latex]
Simplify each radical by identifying perfect squares.
[latex]2\sqrt{4\cdot 10}+\sqrt{25\cdot 10}[/latex]
Use the multiplication property of radicals.
[latex]=2\sqrt{4}\sqrt{10}+\sqrt{25}\sqrt{10}[/latex]
Simplify.
[latex]=2\cdot 2 \sqrt{10}+5 \sqrt{10}[/latex]
Add.
[latex]=4\sqrt{10}+5\sqrt{10}[/latex]
[latex]=9\sqrt{10}[/latex]
Try It
Simplify. [latex]5\sqrt{28}-\sqrt{63}[/latex]
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.
The Product and Quotient Rules for Square Roots
Given that a and b are nonnegative real numbers, [latex]\sqrt{a\cdot{b}}=\sqrt{a}\cdot\sqrt{b}[/latex] and [latex]\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}.[/latex]
We can use these rules in reverse to multiply and divide square roots.
Example
Simplify. [latex]\sqrt{18}\cdot \sqrt{16}[/latex]
Use the rule [latex]\sqrt{a}\cdot \sqrt{b}=\sqrt{ab}[/latex] to multiply the radicands.
[latex]\begin{array}{r}\sqrt{18\cdot 16}\\\\\sqrt{288}\end{array}[/latex]
Look for perfect squares in the radicand, and rewrite the radicand as the product of two factors.
[latex]\sqrt{144\cdot 2}[/latex]
Identify perfect squares.
[latex]\sqrt{{{(12)}^{2}}\cdot 2}[/latex]
Rewrite as the product of two radicals.
[latex]\sqrt{{{(12)}^{2}}}\cdot \sqrt{2}[/latex]
Simplify, using [latex]\sqrt{{{x}^{2}}}=\large\left| \normalsize x \large\right|[/latex].
[latex]12\sqrt{2}[/latex]
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.
Example
Simplify [latex]\sqrt{18}\cdot \sqrt{16}[/latex]
Look for perfect squares in each radicand, and rewrite as the product of two factors.
[latex]\sqrt{9\cdot 2}\cdot \sqrt{4\cdot 4}[/latex]
Rewrite as the product of radicals.
[latex]\sqrt{9}\cdot \sqrt{2}\cdot \sqrt{16}[/latex]
Simplify.
[latex]3\cdot \sqrt{2}\cdot 4[/latex]
Multiply using the commutative and associative properties of multiplication.
[latex](3\cdot 4)\cdot\sqrt{2}[/latex]
[latex]12\sqrt{2}[/latex]
Answer
[latex]\sqrt{18}\cdot \sqrt{16}=12\sqrt{2}[/latex]
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.
Try It
Example
Simplify. [latex]\sqrt{\frac{48}{25}}[/latex]
Use the quotient rule [latex]\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}[/latex] to create two radicals; one in the numerator and one in the denominator.
[latex]\frac{\sqrt{48}}{\sqrt{25}}[/latex]
Simplify each radical. Look for perfect square factors in the radicand, and rewrite the radicand as a product of factors.
[latex]\frac{\sqrt{16\cdot 3}}{\sqrt{25}}[/latex]
Use the product rule for square roots on the numerator.
[latex]\frac{\sqrt{16}\cdot \sqrt{3}}{\sqrt{25}}[/latex]
Simplify the square roots.
[latex]\frac{4\sqrt{3}}{5}[/latex]
Answer
[latex]\sqrt{\frac{48}{25}}=\frac{4\sqrt{3}}{5}[/latex]
Try It
Simplify. [latex]\sqrt{\frac{27}{121}}[/latex]
Example
Simplify. [latex]\sqrt{\frac{640}{40}}[/latex]
Rewrite using the Quotient Rule for square roots.
[latex]\frac{\sqrt{640}}{\sqrt{40}}[/latex]
Simplify each radical. Look for perfect squares in the radicand, and rewrite the radicand as a product of factors.
[latex]\frac{\sqrt{64\cdot 10}}{\sqrt{4\cdot 10}}[/latex]
Rewrite using the Product Rule for square roots.
[latex]\frac{\sqrt{64}\cdot \sqrt{10}}{\sqrt{4}\cdot \sqrt{10}}[/latex]
Take the square roots of the perfect square factors.
[latex]\frac{8\sqrt{10}}{2\sqrt{10}}[/latex]
Simplify the fraction by looking for common factors in the numerator and denominator.
[latex]\frac{4\cdot \color{red}{2}\cdot\color{blue}{\sqrt{10}}}{\color{red}{2}\cdot\color{blue}{\sqrt{10}}}[/latex]
Simplify by cancelling common factors.
[latex]4[/latex]
Answer.
[latex]\sqrt{\frac{640}{40}}=4[/latex]
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.
Example
Simplify. [latex]\sqrt{\frac{640}{40}}[/latex]
Simplify the fraction by looking for common factors on the numerator and denominator.
[latex]\sqrt{\frac{64\cdot 10}{4\cdot 10}}[/latex]
[latex]\sqrt{\frac{16\cdot \color{red}{4}\cdot \color{blue}{10}}{\color{red}{4}\cdot\color{blue}{10}}}[/latex]
Simplify the fraction by cancelling common factors.
[latex]\sqrt{\frac{16}{1}}[/latex]
And since 16 is a perfect square we can take the square root.
[latex]\sqrt{16}=4[/latex]
Answer.
[latex]\sqrt\frac{640}{{40}}=4[/latex].
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].
Example
Rationalize the denominator.
[latex]\frac{-6\sqrt{6}}{\sqrt{3}}[/latex]
The denominator of this fraction is [latex]\sqrt{3}[/latex], which is irrational. To make it into a rational number, we want to turn it into a perfect square [latex]9[/latex] by multiplying it by [latex]\sqrt{3}[/latex]. This means we need to multiply the entire fraction by [latex]\frac{\sqrt{3}}{\sqrt{3}}[/latex] to make an equivalent fraction.
[latex]\begin{align} &= \frac{-6\sqrt{6}}{\sqrt{3}}\cdot\color{blue}{\frac{\sqrt{3}}{\sqrt{3}}}\\ &= \frac{ -6\cdot\sqrt{6\cdot\color{blue}{3}}}{\sqrt{3\cdot\color{blue}{3}}}\;\;\; \;\;\;\text{multiply under the radical}\\ &= \frac{-6\cdot\sqrt{18}}{3}\;\;\;\;\;\; \text{find a perfect square factor under the radical}\\ &= \frac{-6\cdot\sqrt{9\cdot 2}}{3}\;\;\;\;\;\;\text{split up the radical}\\ &=\frac{-6\cdot\sqrt{9}\cdot \sqrt{2}}{3}\;\;\;\;\;\;\text{take the square root}\\ &= \frac{-6\cdot\color{blue}{3}\cdot\sqrt{2}}{\color{blue}{3}}\;\;\;\;\;\;\text{cancel common factors}\\ &= -6\sqrt{2}\end{align}[/latex]
Answer
[latex]\frac{-6\sqrt{6}}{\sqrt{3}}=-6\sqrt{2}[/latex]
Try It
Rationalize the denominator.
[latex]\frac{2\sqrt{3}}{\sqrt{5}}[/latex]
Example
Rationalize the denominator.
[latex]\frac{4\sqrt{3}}{\sqrt{8}}[/latex]
To turn [latex]8[/latex] into a perfect square, we can either multiply by [latex]8[/latex] to get [latex]64[/latex], or multiply by [latex]2[/latex] to get a smaller [latex]16[/latex]. Either way we will get the same answer.
Multiplying by [latex]\frac{\sqrt{8}}{\sqrt{8}}[/latex]:
[latex]\begin{align} &= \frac{4\sqrt{3}}{\sqrt{8}}\cdot\color{blue}{\frac{\sqrt{8}}{\sqrt{8}}}\\ &= \frac{4\sqrt{3\cdot\color{blue}{8}}}{\sqrt{8\cdot\color{blue}{8}}}\;\;\;\text{multiply under the radicals}\\ &= \frac{4\cdot\sqrt{24}}{\sqrt{\color{blue}{64}}}\;\;\;\text{take the square root}\\&= \frac{4\cdot\sqrt{\color{blue}{4\cdot 6}}}{\color{blue}{8}}\;\;\;\text{look for perfect squares under the radical}\\&= \frac{4\cdot\color{blue}{\sqrt{4}\cdot\sqrt{6}}}{8}\;\;\;\text{split up the radical}\\&= \frac{4\cdot\color{blue}{2}\cdot\sqrt{6}}{8}\;\;\;\text{take the square root}\\ &= \frac{\color{blue}{8}\sqrt{6}}{\color{blue}{8}}\;\;\;\text{cancel common factors}\\ &= \sqrt{6}\end{align}[/latex]
Multiplying by [latex]\frac{\sqrt{2}}{\sqrt{2}}[/latex]:
[latex]\begin{align} &= \frac{4\sqrt{3}}{\sqrt{8}}\cdot\color{blue}{\frac{\sqrt{2}}{\sqrt{2}}}\\ &= \frac{4\sqrt{3\cdot\color{blue}{2}}}{\sqrt{8\cdot\color{blue}{2}}}\;\;\;\text{multiply under the radicals}\\ &= \frac{4\cdot\sqrt{6}}{\sqrt{\color{blue}{16}}}\;\;\;\text{take the square root}\\ &= \frac{\color{blue}{4}\sqrt{6}}{\color{blue}{4}}\;\;\;\text{cancel common factors}\\ &= \sqrt{6}\end{align}[/latex]
Although we get the same answer, it is more efficient to use the smallest number that will turn the radicand in the denominator into a perfect square.
Try It
Rationalize the denominator.
[latex]\frac{7\sqrt{6}}{\sqrt{27}}[/latex]
Example
Rationalize the denominator.
[latex]\frac{-3\sqrt{6}}{\sqrt{40}}[/latex]
The denominator can be simplified before rationalizing.
[latex]\begin{align} &=\frac{-3\sqrt{6}}{\sqrt{\color{blue}{4}\cdot 10}}\;\;\;\;\;\text{find a perfect square factor under the radical} \\ &= \frac{-3\sqrt{6}}{\color{blue}{\sqrt{4}\cdot \sqrt{10}}}\;\;\;\;\;\text{split up the radical} \\ &= \frac{-3\sqrt{6}}{\color{blue}{2}\sqrt{10}}\;\;\;\;\;\text{take the square root} \\ &= \frac{-3\sqrt{6}}{2\sqrt{10}}\cdot\color{blue}{\frac{\sqrt{10}}{\sqrt{10}}}\;\;\;\;\;\text{multiply by 1 to make a perfect square on the denominator}\\ &= \frac{-3\sqrt{\color{blue}{6\cdot 10}}}{2\sqrt{\color{blue}{10\cdot 10}}}\;\;\;\;\;\text{simplify under the radicals} \\ &= \frac{-3\sqrt{\color{blue}{60}}}{2\color{blue}{\sqrt{100}}}\;\;\;\;\;\text{find perfect square factors; take the square root} \\ &= \frac{-3\color{blue}{\sqrt{4\cdot 15}}}{2\cdot\color{blue}{10}}\;\;\;\;\;\text{split up the radical} \\ &= \frac{-3\color{blue}{\sqrt{4}\cdot \sqrt{15}}}{2\cdot\color{blue}{10}}\;\;\;\;\;\text{take the square root} \\ &= \frac{-3\cdot\color{blue}{2} \sqrt{15}}{\color{blue}{2}\cdot 10}\;\;\;\;\;\text{cancel common factors} \\ &= \frac{-3\sqrt{15}}{10} \end{align}[/latex]
Try It
Rationalize the denominator.
[latex]\frac{-4\sqrt{2}}{\sqrt{20}}[/latex]
