High Impact Tutoring Built By Math Experts
Personalized standards-aligned one-on-one math tutoring for schools and districts
Here we will learn about radicals, including simplifying radicals, adding and subtracting radicals, multiplying radicals, dividing radicals and rationalizing radicals.
Students are first exposed to a radical, or square root in th grade when taking the square root of numbers. However, they greatly expand that knowledge in algebra class.
Radicals (or sometimes referred to as surds) are represented by and are used to calculate the square root or the nth root of numbers and expressions. Expressions with are called radical expressions.
For example, because is a perfect square number. Perfect square numbers have whole number roots.
However, is an irrational number because it’s a non-terminating, non-repeating decimal that cannot be expressed as a fraction.
Use this quiz to check your grade 6 to 8 students’ understanding of algebra. 10+ questions with answers covering a range of 6th and 8th grade algebra topics to identify areas of strength and support!
DOWNLOAD FREEUse this quiz to check your grade 6 to 8 students’ understanding of algebra. 10+ questions with answers covering a range of 6th and 8th grade algebra topics to identify areas of strength and support!
DOWNLOAD FREEYou can simplify the number under the radical to an exact calculation. For example, let’s apply the Pythagorean Theorem to find the missing length of the triangle.
Notice how is not a perfect square number. One way to find the answer is to type into a calculator.
Another strategy is to simplify the number under the radical. That means to find the largest square root factor of
Here are some perfect square numbers:
is the largest perfect square factor of
simplified radical expression or exact form
You can also simplify radical expressions that contain variables.
For example, let’s simplify the expression,
is a perfect square number so that is simple enough to do. However, how do you simplify Variables raised to even exponents are considered perfect square expressions.
If you notice, does not have an even exponent. However, you can break it apart so that it does.
simplified is
Step-by-step guide: How to simplify radicals
In order to add and subtract radical expressions, the number or expression under the radical symbol must be the same.
For example, can be subtracted because both radicals are identical.
So,
Do not add or subtract the number under the radical. It remains the same.
If the number or expression under the radical is not the same, you must use strategies for simplifying the radical in order to add or subtract them.
For example, cannot be added the way it is written because the radicals do not match.
In this case, the radicals simplify to be the same. So, they can now be added.
Step-by-step guide: Adding radicals
Step-by-step guide: Subtracting radical
Unlike addition and subtraction, the radicals do not have to match when multiplying or dividing. You can also multiply and divide the numbers under the radicals.
For example, The numbers outside the radicals multiply together and the numbers under the radicals multiply together.
Always remember to look at the final product to see if it can be simplified. In this case, the product cannot be simplified any further because there is not a perfect square factor of
Let’s look at another example.
When dividing, the numbers outside of the radicals can divide one another and the numbers inside the radicals can divide one another. In this case, notice how the quotient can be simplified.
Step-by-step guide: Multiplying radicals
Step-by-step guide: Dividing radicals
Expressions that have radicals in the denominator must be rationalized. This means that an equivalent expression has to be created so that it does not have a radical in the denominator.
Let’s look at the expression notice how does not divide evenly into
So, division does not work in this case. The radical cannot remain in the denominator so there is another strategy you can use to rationalize the expression.
Multiply the numerator and denominator by the radical in the denominator, which is The goal is to find an equivalent expression that does not have a radical in the denominator.
Multiplying the numerator and denominator by is like multiplying the expression by which is
the radical is no longer in the denominator
However, the expression can be simplified further. Notice how the in the numerator and the in the denominator can be simplified.
So,
Step-by-step guide: Rationalize the denominator
How does this relate to high school math?
There are lots of different ways we can calculate with radicals. We can:
Let’s have a look at each one in more detail.
Simplify the expression:
Square numbers are
The largest square factor of is because
2Rewrite the radical as a product of the square factor(s) and the other factors.
3Simplify the radical expression.
is a perfect square so take the square root of to simplify
simplifies to be
Simplify the expression:
Find the largest square factor(s) of the term under the root.
is a perfect square and is the largest square factor of
Rewrite the radical as a product of the square factor(s) and the other factors.
Simplify the radical expression.
simplifies to be
Add the expressions:
If the radicals are not alike, simplify them to be the same.
and are not the same.
cannot be simplified.
can be simplified because is a perfect square factor of
Rewrite the expressions to have like radicals.
Add or subtract the expressions.
Subtract the expressions:
If the radicals are not alike, simplify them to be the same.
and are not the same.
can be simplified because is a perfect square factor of
cannot be simplified
Rewrite the expressions to have like radicals.
Add or subtract the expressions.
Divide the radical expression.
Divide the numbers in the radicals and the numbers outside the radicals.
In this case, you are going to divide and
So, and
Simplify radicals that can be simplified and write the new expression.
cannot be simplified.
can be simplified because has a perfect square factor which is
simplifies to be
In this example, you can go even further to simplify because the radicals in both terms match. So, or
Multiply the radical expression.
Multiply the numbers in the radicals and the numbers outside the radicals.
In this case, you will have to use the distributive property.
Simplify radicals that can be simplified and write the new expression.
cannot be simplified and can be simplified because has a perfect square factor which is
Simplify the radical expression.
Multiply the numerator and the denominator of the expression by the radical in the denominator.
The radical in the denominator is
So,
Simplify the expression completely.
can be simplified further because has a perfect square factor which is So, which simplifies to be
can then be simplified further because is a factor of
is simplified completely.
Simplify the expression by rationalizing the denominator.
Write the conjugate.
The conjugate of is
Multiply the numerator and the denominator by the conjugate.
Simplify the expression completely.
can be simplified because is a factor of and
1. Simplify the radical expression fully.
Look for the largest perfect square factor of
is a perfect square number and a factor of
So,
2. Subtract the radical expression.
Simplify the radicals so that the radicals of both terms match.
can be simplified. Rewrite
Both radicals match so the terms can be subtracted.
3. Multiply the radical expressions and simplify the answer fully.
Multiply the numbers inside the radicals and outside the radicals.
can be simplified because has a perfect square factor which is
So,
4. Divide the radical expression and simplify the answer fully.
Divide the number under the radicals.
Simplify the radical because has a perfect square factor of
5. Simplify the expression.
can be simplified because is a perfect square number.
can be rewritten as and is a perfect square term.
So,
6. Expand the expression and simplify.
Use the distributive property to multiply.
can be simplified because has a perfect square factor which is
So,
So the final answer is
7. Rationalize the denominator of the radical expression.
Multiply the numerator and denominator by
A rational exponent is a fractional exponent that represents a radical expression. For example, the irrational number, can be written as
When you get into algebra you will learn how to work with the square root of negative numbers. The square root of negative numbers produce imaginary numbers.
The absolute value of a complex number is in the form of
You can take the cube root of perfect cube numbers similarly to the way you can take the square root of perfect square numbers.
For example, Notice how the index of the radical is that’s because you are taking a cube root and not a square root.
Radical equations are equations that have radical expressions. Solving radical equations requires undoing the radical. For example, squaring a square root expression undoes the square root and cubing a cube root expression undoes the cube root.
In algebra and algebra you will analyze the graph of rational expressions/equations, logarithm equations, polynomial equations, quadratic equations, and linear equations. You will learn about their features and what their graphs look like.
At Third Space Learning, we specialize in helping teachers and school leaders to provide personalized math support for more of their students through high-quality, online one-on-one math tutoring delivered by subject experts.
Each week, our tutors support thousands of students who are at risk of not meeting their grade-level expectations, and help accelerate their progress and boost their confidence.
Find out how we can help your students achieve success with our math tutoring programs.
Prepare for math tests in your state with these 3rd Grade to 8th Grade practice assessments for Common Core and state equivalents.
Get your 6 multiple choice practice tests with detailed answers to support test prep, created by US math teachers for US math teachers!
Notifications