Ready to move past simplifying radicals and rationalizing the denominator? Let’s get started! In this article, we will cover basic operations with radicals. This includes adding, subtracting, multiplying, and dividing radicals. We’ll discuss the basic rules for each operation and then work through specific examples. Here we go!
What We Review
How to add radicals
The key rule to remember is that adding radicals requires the same radicand and index for all terms. This principle is similar to combining like terms in algebraic expressions.
A general guideline for combining radicals is:
\sqrt[a]{b} + \sqrt[a]{b} = 2\sqrt[a]{b}
This indicates that addition or subtraction is only allowed when the radicals share both their index and radicand.
Adding and subtracting radicals follow the same guidelines, so let’s look at examples of both.
Example 1
Simplify: \sqrt{18} + 2\sqrt{2} + 3\sqrt{18}
Adding square roots requires us to simplify first. \sqrt{18} simplifies to 3\sqrt{2} because 18 can be expressed as 9 * 2, and the square root of 9 is 3.
Thus, the expression becomes: 3\sqrt{2} + 2\sqrt{2} + 9\sqrt{2}
Since all terms are like radicals (\sqrt{2}), they can be combined to: 14\sqrt{2}
Therefore, the simplified form of \sqrt{18} + 2\sqrt{2} + 3\sqrt{18} is 14\sqrt{2}.
Example 2
Now let’s try an example with subtracting radicals.
Consider the subtraction: 5\sqrt{50} - \sqrt{8}
Simplify 5\sqrt{50} to 25\sqrt{2}, because 50 can be broken down to 25 * 2, and the square root of 25 is 5.
Simplify \sqrt{8} to 2\sqrt{2}, because 8 can be expressed as 4 * 2, and the square root of 4 is 2.
Subtracting the like radicals yields: 25\sqrt{2} - 2\sqrt{2} = 23\sqrt{2}
Thus, 5\sqrt{50} - \sqrt{8} simplifies to 23\sqrt{2}.
For more examples on adding and subtracting radical expressions, check out this video:
How to multiply radicals
Multiplying radicals introduces a new level of interaction between radical expressions, yet adheres to the foundational principles of algebra. The process is straightforward: when multiplying radicals, you multiply the radicands together while keeping them under the same radical sign, provided the radicals have the same index.
A basic rule for multiplying radicals is:
\sqrt[a]{b} \cdot \sqrt[a]{c} = \sqrt[a]{b \cdot c}
This indicates that the product of two radicals is a radical of the product of their radicands, assuming identical indices.
Example 1
Multiply: \sqrt{12} \cdot \sqrt{3}
First, apply the rule for multiplying radicals: \sqrt{12 \cdot 3}
Simplify the radicand by multiplying: \sqrt{36}
Since 36 is a perfect square, it simplifies to: 6
Therefore, the product of \sqrt{12} \cdot \sqrt{3} is 6.
Example 2
Multiply: 2\sqrt{5} \cdot 3\sqrt{10}
First, multiply the coefficients (outside the radicals) and the radicands (inside the radicals) separately: 2 \cdot 3\sqrt{5 \cdot 10}
Simplify the multiplication: 6\sqrt{50}
Further simplify the radical: 6\sqrt{25 \cdot 2}
Since \sqrt{25} is 5, this becomes: 6 \cdot 5\sqrt{2}
Therefore, the simplified form of 2\sqrt{5} \cdot 3\sqrt{10} is 30\sqrt{2}.
How to divide radicals
Dividing with square roots involves a process that might initially seem intricate, but it follows logical steps that are easy to grasp once understood. When dividing radicals, especially those with square roots, the goal is to simplify the division to a form where no radical appears in the denominator.
A fundamental approach for dividing radicals is:
\frac{\sqrt[a]{b}}{\sqrt[a]{c}} = \sqrt[a]{\frac{b}{c}}
This formula demonstrates that dividing two radicals with the same index is equivalent to taking the square root of the quotient of their radicands.
Example 1
Divide: \frac{\sqrt{48}}{\sqrt{3}}
Apply the rule for dividing radicals: \sqrt{\frac{48}{3}}
Simplify the division inside the radical: \sqrt{16}
Since 16 is a perfect square, it simplifies to: 4
Therefore, the quotient of \frac{\sqrt{48}}{\sqrt{3}} is 4.
Example 2
Divide: \frac{4\sqrt{18}}{2\sqrt{2}}
First, simplify each radical: \frac{4\sqrt{9 \cdot 2}}{2\sqrt{2}}
This simplifies to: \frac{4 \cdot 3\sqrt{2}}{2\sqrt{2}}
Simplify further by dividing the coefficients and radicals: \frac{12}{2}\cdot \frac{\sqrt{2}}{\sqrt{2}}
Therefore, the simplified form of \frac{4\sqrt{18}}{2\sqrt{2}} is 6.
For more practice on multiplying and dividing radicals, check out this video here:
Return to the Table of Contents
Multiple operations with radicals
When engaging with operations with radicals that encompass a mix of addition, subtraction, multiplication, and division, it’s essential to approach these radical expressions with a clear strategy. Handling multiple operations requires an understanding of the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)) alongside the specific rules governing operations with radicals.
Example 1
Simplify: \sqrt{32} + 2\sqrt{2} - \sqrt{18} \cdot \frac{\sqrt{8}}{\sqrt{2}}
This example is carefully chosen to require the application of all the operations with radicals we’ve discussed. Let’s break it down step by step:
Simplify the Radicals:
\sqrt{32} can be simplified to 4\sqrt{2} because 32 = 16 \cdot 2 and \sqrt{16} = 4. \sqrt{18} simplifies to 3\sqrt{2} because 18 = 9 \cdot 2 and \sqrt{9} = 3. \sqrt{8} simplifies to 2\sqrt{2} because 8 = 4 \cdot 2 and \sqrt{4} = 2.Now we have the following radical expression: 4\sqrt{2}+ 2\sqrt{2} - 3\sqrt{2} \cdot \frac{ \sqrt{8}}{\sqrt{2} }
Simplify again by dividing the radicals: 4\sqrt{2}+ 2\sqrt{2} - 3\sqrt{2} \cdot \sqrt{4}
Since \sqrt{4}=2, we can multiply: 4\sqrt{2}+ 2\sqrt{2} - 6\sqrt{2}
Adding the radicals gives us: 6\sqrt{2}- 6\sqrt{2}
Finally, subtracting the radicals gives us: 0\sqrt{2}=0
Return to the Table of Contents
Conclusion: Operations with Radicals
We’ve covered a lot in this guide about working with radicals, from adding and subtracting to multiplying and dividing them. By breaking down each step and looking at examples, we’ve seen it’s not as tough as it might seem at first. Now that you know how to handle these math problems, you’re all set to tackle them on your own with a lot more confidence.
