Operations with Radicals (Square Roots)
In order to find EXACT (not decimal) solutions to quadratic equations, we need to learn how to simplify square roots. And while we are working with square roots, it’s not a bad idea to review other operations with radicals, including multiplication, division, addition and subtraction.
Key Vocabulary
The expression \(\sqrt{a}\) is called a radical. The symbol \(\sqrt{\;\;}\) is called a radical sign and the number \(a\) under the radical sign is called the radicand of the expression. When simplifying square roots, we write the principal square root (which is positive). For example, \(\sqrt{36}=6\) even though both \(6^2 = 36\) and \((-6)^2 = 36\). However, when you solve an equation like \(x^2=36\), you will always get two solutions (more on this in future lessons).
In order to find EXACT (not decimal) solutions to quadratic equations, we need to learn how to simplify square roots. And while we are working with square roots, it’s not a bad idea to review other operations with radicals, including multiplication, division, addition and subtraction.
Key Vocabulary
The expression \(\sqrt{a}\) is called a radical. The symbol \(\sqrt{\;\;}\) is called a radical sign and the number \(a\) under the radical sign is called the radicand of the expression. When simplifying square roots, we write the principal square root (which is positive). For example, \(\sqrt{36}=6\) even though both \(6^2 = 36\) and \((-6)^2 = 36\). However, when you solve an equation like \(x^2=36\), you will always get two solutions (more on this in future lessons).
Properties of Square Roots (for \(a>0\ and\ b>0\)): Product Property: \(\sqrt{a\cdot b}=\sqrt{a}\cdot\sqrt{b}\) Quotient Property: \(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\) Power Property: \(\sqrt{a^2}=\left(\sqrt{a}\right)^2=a\) |
Examples of Square Roots (for \(a>0\ and\ b>0\)): Product Property: \( \sqrt{48}=\sqrt{16\cdot3}=\sqrt{16}\cdot\sqrt{3}=4\sqrt{3}\) Quotient Property: \(\sqrt{\frac{7}{16}}=\frac{\sqrt{7}}{\sqrt{16}}=\frac{\sqrt{7}}{4}\) Quotient Property: \(\sqrt{25}=\sqrt{5^2}=\left(\sqrt{5}\right)^2=5\) |
You can use these three properties (as illustrated in the above examples) to simplify square roots. A square root is fully simplified when the following three conditions are met:
Ideally, you should be able to find the largest perfect square factor of any radicand using mental math, but there are a few methods that will work if you are unable to find the correct factor quickly.
One method is to find the largest perfect square factor of the radicand then apply the product property. Another method is to create a prime factor tree to write the radicand as a power of primes then apply the power property. Both methods are explained in the video examples, so you will probably want to watch the videos!
Example 1: Simplify \(\sqrt{72}\).
- There are no perfect square factors (other than \(1\)) in the radicand.
- There is no radical in the denominator of a fraction.
- The radicand is not a fraction.
Ideally, you should be able to find the largest perfect square factor of any radicand using mental math, but there are a few methods that will work if you are unable to find the correct factor quickly.
One method is to find the largest perfect square factor of the radicand then apply the product property. Another method is to create a prime factor tree to write the radicand as a power of primes then apply the power property. Both methods are explained in the video examples, so you will probably want to watch the videos!
Example 1: Simplify \(\sqrt{72}\).
Example 2: Simplify \(\sqrt{18}\).
Sometimes you need to apply the product property before you can simplify the square root, as illustrated below.
Example 3: Simplify \(\sqrt{10}\cdot\sqrt{15}\).
\(\begin{align}&\sqrt{10}\cdot\sqrt{15}\ & \ \ \ & \text{1) Rewrite original problem.}\\&\sqrt{10\cdot15}\ & \ \ \ & \text{2) Apply the product property.}\\&\sqrt{150}\ & \ \ \ & \text{3) Multiply.}\\&\sqrt{25\cdot6}\ & \ \ \ & \text{4) Find the largest perfect square factor.}\\&\sqrt{25}\cdot\sqrt{6}\ & \ \ & \text{5) Apply the product property.}\\&5\sqrt{6}\ & \ \ \ & \text{6) Simplify the square root of the perfect square.}\end{align}\)
Example 4: Simplify \(\large\sqrt{\frac{125}{9}}\).
\(\begin{align}\sqrt{\frac{125}{9}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & \text{1) Rewrite original problem.}\\\frac{\sqrt{125}}{\sqrt{9}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & \text{2) Apply the quotient property.}\\\frac{\sqrt{25\cdot5}}{3}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & \text{3) Find the largest perfect square factor in the numerator; simplify the square root in the denominator.}\\\frac{\sqrt{25}\sqrt{5}}{3}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & \text{4) Apply the product property in the numerator.}\\
\frac{5\sqrt{5}}{3}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & \text{5) Simplify the square root of the perfect square.}\end{align}\)
\frac{5\sqrt{5}}{3}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & \text{5) Simplify the square root of the perfect square.}\end{align}\)
A few things to note with division of radicals:
- It is sometimes easier to reduce the fraction under the radical sign before you apply the quotient rule.
- You cannot divide a number under the radical sign with a number not under the radical sign.
- You should reduce all fractions whenever possible.
So what happens when the denominator of the fraction is not a perfect square? What if the denominator has the form \(\sqrt{b}\ \text{or}\ \ a+\sqrt{b}\ \text{or}\ \ a-\sqrt{b}\)? In these cases, we rationalize the denominator in order to eliminate the radical in the denominator. The table below illustrates what the numerator and denominator of the fraction need to be multiplied by in order to rationalize the denominator.
If the denominator is... |
Multiply the numerator and denominator by... |
\(\sqrt{b}\) |
\(\sqrt b\) |
\(a + \sqrt{b}\) |
\(a - \sqrt b\) |
\(a - \sqrt b\) |
\(a+\sqrt b\) |
The expressions \(a+\sqrt{b}\) and \(a-\sqrt{b}\) are called conjugates of each other. If you multiply the two expressions together, you will always get a rational number. The conjugate is formed by changing the sign between the rational number and the radical.
Example 5: Simplify \(\large\frac{48}{\sqrt{3}}\).
\(\begin{align}\frac{48}{\sqrt{3}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & \text{1) Rewrite original problem.}\\\frac{48}{\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & \text{2) Multiply numerator and denominator by the square root in the denominator.}\\\frac{48\sqrt{3}}{3}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & \text{3) Apply the power rule in the denominator.}\\16\sqrt{3}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & \text{4) Reduce the fraction.}\end{align}\)
Example 6: Simplify \(\large\sqrt{\frac{7}{2}}\).
Example 7: Simplify \(\large\frac{5}{3-\sqrt{2}}\).
Last, we will look at addition and subtraction of radicals. (I know--lots of content here).
To add or subtract square roots, the radicands must be equal. Do not EVER add or subtract the numbers under the square root sign!
Quick Check
Without a calculator, decide whether each statement is true or false.
1) \(\sqrt{4}+\sqrt{9}=\sqrt{13}\)
2) \(\sqrt{4}+\sqrt{4}=\sqrt{8}\)
3) \(\sqrt{4}+\sqrt{4}=2\sqrt{4}\)
4) \(\sqrt{16}-\sqrt{4}=\sqrt{12}\)
5) \(5\sqrt{3}+\sqrt{3}-2\sqrt{3}=4\sqrt{3}\)
Quick Check Solutions
Without a calculator, decide whether each statement is true or false.
1) \(\sqrt{4}+\sqrt{9}=\sqrt{13}\)
2) \(\sqrt{4}+\sqrt{4}=\sqrt{8}\)
3) \(\sqrt{4}+\sqrt{4}=2\sqrt{4}\)
4) \(\sqrt{16}-\sqrt{4}=\sqrt{12}\)
5) \(5\sqrt{3}+\sqrt{3}-2\sqrt{3}=4\sqrt{3}\)
Quick Check Solutions
To add or subtract square roots:
- Simplify each radical, if necessary.
- If the radicands are equal, add or subtract the coefficients (the numbers outside of the radical). It’s like combining like terms but instead of having a variable, you have a square root.
Example 8: Simplify \(3\sqrt{72}+\sqrt{18}\).
\(\begin{align}3\sqrt{72}&+\sqrt{18}\ & \ \ \ & \text{1) Rewrite original problem.}\\3\sqrt{36\cdot2}&+\sqrt{9\cdot2}\ & \ \ \ & \text{2) Find largest perfect square factors.}\\3\sqrt{36}\cdot\sqrt{2}&+\sqrt{9}\cdot\sqrt{2}\ & \ \ \ & \text{3) Apply product property.}\\3\cdot6\sqrt{2}&+3\sqrt{2}\ & \ \ \ & \text{4) Simplify each square root.}\\18\sqrt{2}&+3\sqrt{2}\ & \ \ \ & \text{5) Multiply.}\\&21\sqrt{2}\ \ & \ \ & \text{6) Add the coefficients.}\end{align}\)
Example 9: Simplify \(-5\sqrt{27}-8\sqrt{12}+4\sqrt{8}\).