Are you tired of feeling confused or overwhelmed when it comes to radicals in math? Fear not, because in this blog post, we’ll break down everything you need to know about simplifying radicals with easy-to-follow examples.
This detailed post is all about helping you understand how to simplify radicals step-by-step with examples, and we’ll also cover how to simplify radicals with variables. Whether you’re a student or a teacher, this post will be a useful guide to help you master simplifying radicals. So, let’s get started!
What We Review
What is a radical in math?
In math, a radical is a mathematical symbol representing the root of a number or expression, including square and cube roots.
The radical symbol looks like a check mark or the letter “v”. The symbol is placed in front of the expression being rooted, and the expression inside the radical is called the radicand. For example, the square root of 25 is written as \sqrt{25}, where 25 is the radicand.
In radicals, the index refers to the number above the radical symbol that specifies which root is being taken. For example, in the expression \sqrt[3]{64}, the index is 3 because a cube root is being taken, and the radicand is 64, as shown below:
The index is typically omitted when taking a square root, as the square root is assumed to be the default root when no index is specified.
What does “root” mean in math?
Before we can start to simplify radicals, we need to understand what finding a root means.
“Rooting” a number means finding the number that, when multiplied by itself a certain number of times, results in the original number.
For example, the square root of 25 is 5, because 5 multiplied by itself equals 25.
Similarly, the cube root of 8 is 2, because 2 multiplied by itself three times equals 8.
The root of a number is represented using a radical symbol, such as \sqrt{} for the square root or \sqrt[3]{} for the cube root.
What does simplifying radicals mean?
Before jumping straight into examples, let’s clarify this concept of “simplifying”.
Simplifying a radical means expressing it in its simplest form, where the radicand has no perfect square factors remaining inside the radical symbol.
A perfect square factor is a number that can be expressed as the product of two identical integers. For example, 4 is a perfect square because it can be expressed as 2 \times 2 . Similarly, 9 is a perfect square because it can be expressed as 3 \times 3 . When simplifying a radical, we look for perfect square factors in the radicand (the number inside the radical symbol), and factor them out of the radical symbol.
Simplifying a radical leaves us with a simplified form of the radical, with no perfect square factors remaining inside the radical symbol.
For example, \sqrt{12} can be simplified to 2\sqrt{3}, since 12 can be factored as 4 \times 3 , and 4 is a perfect square.
However, \sqrt{7} cannot be simplified any further because 7 has no perfect square factor. Simplifying radicals is an essential skill in solving and simplifying algebraic expressions. We should note that while simplifying radicals generally makes an expression easier to work with, it’s not always necessary to simplify a radical completely. Sometimes, leaving the radical in a slightly more complex form may be more useful, such as when simplifying expressions with variables.
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How to simplify square roots
To simplify a square root, we factor the radicand into its prime factors and group any pairs of identical factors together.
Next, we take the square root of each perfect square factor and multiply them together outside the radical symbol. The remaining factors are left inside the radical.
Examples: simplifying radicals (square roots)
Example 1
Simplify \sqrt{16}
Let’s think about if there are any whole numbers that, when multiplied by themselves, result in 16 .
There is a number that fits this description: 4
Since 16 is a perfect square, we can take the square root of 16 to get 4.
Therefore, \sqrt{16} can be simplified to 4.
Example 2
Simplify \sqrt{75}
No whole number times itself results in 75, so there is no perfect square of 75. We should start to think about any perfect squares that could be factored out of 75.
We can factor the radicand 75 into its prime factors: 75 = 3 \cdot 5^2.
Identify any perfect square factors. In this case, we have a pair of 5’s, a perfect square factor!
We can write the pair of identical factors as a square and take the square root outside of the radical. In this case, we can write 5^2 as (5 \cdot 5), and simplify it as \sqrt{5^2} = 5.
Multiply the perfect square factor outside the radical symbol by any remaining factors inside the radical. In this case, we have the factor 3 inside the radical. We can write the simplified form of \sqrt{75} as 5\sqrt{3}.
Therefore, \sqrt{75} can be simplified to 5\sqrt{3}.
For more examples of how to simplify roots (including cube roots), check out this video:
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Simplifying radical expressions
A radical expression is a numerical or algebraic expression that includes a radical. Simplifying radical expressions involves simplifying the radical by combining like terms and performing any necessary operations, such as addition, subtraction, multiplication, and division.
To simplify a radical expression, we follow the same steps as simplifying square roots: factor the radicand into its prime factors, group any pairs of identical factors, take the square root of each perfect square factor, and multiply them together outside the radical symbol.
However, when we have variables inside the radical, we need to be careful to simplify the expression as much as possible without changing its value. We also need to make sure to follow the order of operations when simplifying an expression with radicals.
Let’s try some examples to see how this works!
Examples: simplifying numerical radical expressions
Below are two detailed examples of simplifying radical expressions with numbers (no variables).
Example 1
Simplify \sqrt{27} - \sqrt{12}
First, we factor the radicand of \sqrt{27}:
\sqrt{27} = \sqrt{3 \cdot 9}
Next, we simplify the perfect square factor of 9: \sqrt{3 \cdot 9} = 3\sqrt{3}.
We can factor the radicand of \sqrt{12}: \sqrt{12} = \sqrt{4 \cdot 3}.
…and then simplify the perfect square factor of 4: \sqrt{4 \cdot 3} = 2\sqrt{3}.
Substitute the simplified forms of each radical into the original expression:
\sqrt{27} - \sqrt{12} = 3\sqrt{3} - 2\sqrt{3}.
..and then combine our like terms:
3\sqrt{3} - 2\sqrt{3} = \sqrt{3}
So, \sqrt{27} - \sqrt{12} simplifies to:
\sqrt{3}
Nice work! Let’s try another one to get the hang of it.
Example 2
Simplify 2\sqrt{18} + 3\sqrt{8}
To start, we can factor the radicand of 2\sqrt{18}:
2\sqrt{18} = 2 \cdot \sqrt{9 \cdot 2}.
Next, simplify the perfect square factor of 9 :
2 \cdot \sqrt{9 \cdot 2} = 2 \cdot 3\sqrt{2} = 6\sqrt{2}
Factor the radicand of 3\sqrt{8}:
3\sqrt{8} = 3 \cdot \sqrt{4 \cdot 2}
Simplify the perfect square factor of 4
3 \cdot \sqrt{4 \cdot 2} = 3 \cdot 2\sqrt{2} = 6\sqrt{2}.
.. and then substitute the simplified forms of each radical into the original expression:
2\sqrt{18} + 3\sqrt{8} = 6\sqrt{2} + 6\sqrt{2}
Combine our lovely like terms:
6\sqrt{2} + 6\sqrt{2} = 12\sqrt{2}
So, 2\sqrt{18} + 3\sqrt{8} simplifies to:
12\sqrt{2}
Examples: simplify radical expressions with variables
Now we’ll extend our learning to show two step-by-step examples of simplifying radical expressions that include variables.
Example 1
Simplify \sqrt{x^4}
We can write x^4 as the square of x^2, so:
x^4 = (x^2)^2
This means that x^4 is a perfect square, so it can be simplified as follows:
\sqrt{x^4} = \sqrt{(x^2)^2} = x^2
Therefore, \sqrt{x^4} simplifies to x^2.
Example 2
Simplify \sqrt{27x^3y^5}
This example is much more difficult than the first example. Fear not: we can do it!
To start, we can break down the radicand into its prime factors:
\sqrt{27x^3y^5} = \sqrt{(3^3 \cdot x^2 \cdot y^4 \cdot y)}
Next, identify any perfect square factors. In this case, there are no perfect square factors.
So, we can start to simplify the expression. Using the product rule of radicals, we can simplify the expression as follows:
\sqrt{27x^3y^5} = \sqrt{3^2 \cdot 3 \cdot x^2 \cdot y^4 \cdot y} = \sqrt{3^2} \cdot \sqrt{3} \cdot \sqrt{x^2} \cdot \sqrt{y^4} \cdot \sqrt{y}
= 3xy^2\sqrt{3y}
Therefore, \sqrt{27x^3y^5} simplifies to:
3xy^2\sqrt{3y}
For more examples of simplifying more complicated radical expressions with variables, watch this wonderful video:
Conclusion: Simplifying Radicals
Knowing how to simplify radicals is an important skill in algebra. Simplifying a radical means ensuring the radicand has no perfect square factors remaining inside the radical symbol.
You can simplify numerical and algebraic expressions with radicals by breaking down the radicand into its prime factors and simplifying perfect square factors. Remembering these steps and practicing with examples can help you become more confident and proficient in simplifying radicals.
