8.3: Simplify Radical Expressions
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By the end of this section, you will be able to:
- Use the Product Property to simplify radical expressions
- Use the Quotient Property to simplify radical expressions
Before you get started, take this readiness quiz.
- Simplify: x9x4.
If you missed this problem, review Example 5.13. - Simplify: y3y11.
If you missed this problem, review Example 5.13. - Simplify: (n2)6.
If you missed this problem, review Example 5.17.
Use the Product Property to Simplify Radical Expressions
We will simplify radical expressions in a way similar to how we simplified fractions. A fraction is simplified if there are no common factors in the numerator and denominator. To simplify a fraction, we look for any common factors in the numerator and denominator.
A radical expression, n√a, is considered simplified if it has no factors of mn. So, to simplify a radical expression, we look for any factors in the radicand that are powers of the index.
For real numbers a and m, and n≥2,
n√a is considered simplified if a has no factors of mn
For example, √5 is considered simplified because there are no perfect square factors in 5. But √12 is not simplified because 12 has a perfect square factor of 4.
Similarly, 3√4 is simplified because there are no perfect cube factors in 4. But 3√24 is not simplified because 24 has a perfect cube factor of 8.
To simplify radical expressions, we will also use some properties of roots. The properties we will use to simplify radical expressions are similar to the properties of exponents. We know that
(ab)n=anbn.
The corresponding of Product Property of Roots says that
n√ab=n√a⋅n√b.
If n√a and n√b are real numbers, and n≥2 is an integer, then
n√ab=n√a⋅n√b and n√a⋅n√b=n√ab
We use the Product Property of Roots to remove all perfect square factors from a square root.
Simplify: √98.
Solution:
Step 1: Find the largest factor in the radicand that is a perfect power of the index. |
We see that 49 is the largest factor of 98 that has a power of 2. |
√98 |
Rewrite the radicand as a product of two factors, using that factor. |
In other words 49 is the largest perfect square factor of 98. 98=49⋅2 Always write the perfect square factor first. |
√49⋅2 |
Step 2: Use the product rule to rewrite the radical as the product of two radicals. | √49⋅√2 | |
Step 3: Simplify the root of the perfect power. | 7√2 |
Simplify: √48
- Answer
-
4√3
Simplify: √45.
- Answer
-
3√5
Notice in the previous example that the simplified form of √98 is 7√2, which is the product of an integer and a square root. We always write the integer in front of the square root.
Be careful to write your integer so that it is not confused with the index. The expression 7√2 is very different from 7√2.
Simplify a Radical Expression Using the Product Property
- Find the largest factor in the radicand that is a perfect power of the index. Rewrite the radicand as a product of two factors, using that factor.
- Use the product rule to rewrite the radical as the product of two radicals.
- Simplify the root of the perfect power.
We will apply this method in the next example. It may be helpful to have a table of perfect squares, cubes, and fourth powers.
Simplify:
- √500
- 3√16
- 4√243
Solution:
a.
√500
Rewrite the radicand as a product using the largest perfect square factor.
√100⋅5
Rewrite the radical as the product of two radicals.
√100⋅√5
Simplify.
10√5
b.
3√16
Rewrite the radicand as a product using the greatest perfect cube factor. 23=8
3√8⋅2
Rewrite the radical as the product of two radicals.
3√8⋅3√2
Simplify.
23√2
c.
4√243
Rewrite the radicand as a product using the greatest perfect fourth power factor. 34=81
4√81⋅3
Rewrite the radical as the product of two radicals.
4√81⋅4√3
Simplify.
34√3
Simplify: a. √288 b. 3√81 c. 4√64
- Answer
-
a. 12√2 b. 33√3 c. 24√4
Simplify: a. √432 b. 3√625 c. 4√729
- Answer
-
a. 12√3 b. 53√5 c. 34√9
The next example is much like the previous examples, but with variables. Don’t forget to use the absolute value signs when taking an even root of an expression with a variable in the radical.
Simplify:
- √x3
- 3√x4
- 4√x7
Solution:
a.
√x3
Rewrite the radicand as a product using the largest perfect square factor.
√x2⋅x
Rewrite the radical as the product of two radicals.
√x2⋅√x
Simplify.
|x|√x
b.
3√x4
Rewrite the radicand as a product using the largest perfect cube factor.
3√x3⋅x
Rewrite the radical as the product of two radicals.
3√x3⋅3√x
Simplify.
x3√x
c.
4√x7
Rewrite the radicand as a product using the greatest perfect fourth power factor.
4√x4⋅x3
Rewrite the radical as the product of two radicals.
4√x4⋅4√x3
Simplify.
|x|4√x3
Simplify: a. √b5 b. 4√y6 c. 3√z5
- Answer
-
a. b2√b b. |y|4√y2 c. z3√z2
Simplify: a. √p9 b. 5√y8 c. 6√q13
- Answer
-
a. p4√p b. p5√p3 c. q26√q
We follow the same procedure when there is a coefficient in the radicand. In the next example, both the constant and the variable have perfect square factors.
Simplify:
- √72n7
- 3√24x7
- 4√80y14
Solution:
a.
√72n7
Rewrite the radicand as a product using the largest perfect square factor.
√36n6⋅2n
Rewrite the radical as the product of two radicals.
√36n6⋅√2n
Simplify.
6|n3|√2n
b.
3√24x7
Rewrite the radicand as a product using perfect cube factors.
3√8x6⋅3x
Rewrite the radical as the product of two radicals.
3√8x6⋅3√3x
Rewrite the first radicand as (2x2)3.
3√(2x2)3⋅3√3x
Simplify.
2x23√3x
c.
4√80y14
Rewrite the radicand as a product using perfect fourth power factors.
4√16y12⋅5y2
Rewrite the radical as the product of two radicals.
4√16y12⋅4√5y2
Rewrite the first radicand as (2y3)4.
4√(2y3)4⋅4√5y2
Simplify.
2|y3|4√5y2
Simplify: a. √32y5 b. 3√54p10 c. 4√64q10
- Answer
-
a. 4y2√2y b. 3p33√2p c. 2q24√4q2
Simplify: a. √75a9 b. 3√128m11 c. 4√162n7
- Answer
-
a. 5a4√3a b. 4m33√2m2 c. 3|n|4√2n3
In the next example, we continue to use the same methods even though there are more than one variable under the radical.
Simplify:
- √63u3v5
- 3√40x4y5
- 4√48x4y7
Solution:
a.
√63u3v5
Rewrite the radicand as a product using the largest perfect square factor.
√9u2v4⋅7uv
Rewrite the radical as the product of two radicals.
√9u2v4⋅√7uv
Rewrite the first radicand as (3uv2)2.
√(3uv2)2⋅√7uv
Simplify.
3|u|v2√7uv
b.
3√40x4y5
Rewrite the radicand as a product using the largest perfect cube factor.
3√8x3y3⋅5xy2
Rewrite the radical as the product of two radicals.
3√8x3y3⋅3√5xy2
Rewrite the first radicand as (2xy)3.
3√(2xy)3⋅3√5xy2
Simplify.
2xy3√5xy2
c.
4√48x4y7
Rewrite the radicand as a product using the largest perfect fourth power factor.
4√16x4y4⋅3y3
Rewrite the radical as the product of two radicals.
4√16x4y4⋅4√3y3
Rewrite the first radicand as (2xy)4.
4√(2xy)4⋅4√3y3
Simplify.
2|xy|4√3y3
Simplify:
- √98a7b5
- 3√56x5y4
- 4√32x5y8
- Answer
-
- 7|a3|b2√2ab
- 2xy3√7x2y
- 2|x|y24√2x
Simplify:
- √180m9n11
- 3√72x6y5
- 4√80x7y4
- Answer
-
- 6m4|n5|√5mn
- 2x2y3√9y2
- 2|xy|4√5x3
Simplify:
- 3√−27
- 4√−16
Solution:
a.
3√−27
Rewrite the radicand as a product using perfect cube factors.
3√(−3)3
Take the cube root.
−3
b.
4√−16
There is no real number n where n4=−16.
Not a real number
Simplify:
- 3√−64
- 4√−81
- Answer
-
- −4
- no real number
Simplify:
- 3√−625
- 4√−324
- Answer
-
- −53√5
- no real number
We have seen how to use the order of operations to simplify some expressions with radicals. In the next example, we have the sum of an integer and a square root. We simplify the square root but cannot add the resulting expression to the integer since one term contains a radical and the other does not. The next example also includes a fraction with a radical in the numerator. Remember that in order to simplify a fraction you need a common factor in the numerator and denominator.
Simplify:
- 3+√32
- 4−√482
Solution:
a.
3+√32
Rewrite the radicand as a product using the largest perfect square factor.
3+√16⋅2
Rewrite the radical as the product of two radicals.
3+√16⋅√2
Simplify.
3+4√2
The terms cannot be added as one has a radical and the other does not. Trying to add an integer and a radical is like trying to add an integer and a variable. They are not like terms!
b.
4−√482
Rewrite the radicand as a product using the largest perfect square factor.
4−√16⋅32
Rewrite the radical as the product of two radicals.
4−√16⋅√32
Simplify.
4−4√32
Factor the common factor from the numerator.
4(1−√3)2
Remove the common factor, 2, from the numerator and denominator.
2⋅2(1−√3)2
Simplify.
2(1−√3)
Simplify:
- 5+√75
- 10−√755
- Answer
-
- 5+5√3
- 2−√3
Simplify:
- 2+√98
- 6−√453
- Answer
-
- 2+7√2
- 2−√5
Use the Quotient Property to Simplify Radical Expressions
Whenever you have to simplify a radical expression, the first step you should take is to determine whether the radicand is a perfect power of the index. If not, check the numerator and denominator for any common factors, and remove them. You may find a fraction in which both the numerator and the denominator are perfect powers of the index.
Simplify:
- √4580
- 3√1654
- 4√580
Solution:
a.
√4580
Simplify inside the radical first. Rewrite showing the common factors of the numerator and denominator.
√5⋅95⋅16
Simplify the fraction by removing common factors.
√916
Simplify. Note (34)2=916.
34
b.
3√1654
Simplify inside the radical first. Rewrite showing the common factors of the numerator and denominator.
3√2⋅82⋅27
Simplify the fraction by removing common factors.
3√827
Simplify. Note (23)3=827.
23
c.
4√580
Simplify inside the radical first. Rewrite showing the common factors of the numerator and denominator.
4√5⋅15⋅16
Simplify the fraction by removing common factors.
4√116
Simplify. Note (12)4=116.
12
Simplify:
- √7548
- 3√54250
- 4√32162
- Answer
-
- 54
- 35
- 23
Simplify:
- √98162
- 3√24375
- 4√4324
- Answer
-
- 79
- 25
- 13
In the last example, our first step was to simplify the fraction under the radical by removing common factors. In the next example we will use the Quotient Property to simplify under the radical. We divide the like bases by subtracting their exponents,
aman=am−n,a≠0
Simplify:
- √m6m4
- 3√a8a5
- 4√a10a2
Solution:
a.
√m6m4
Simplify the fraction inside the radical first. Divide the like bases by subtracting the exponents.
√m2
Simplify.
|m|
b.
3√a8a5
Use the Quotient Property of exponents to simplify the fraction under the radical first.
3√a3
Simplify.
a
c.
4√a10a2
Use the Quotient Property of exponents to simplify the fraction under the radical first.
4√a8
Rewrite the radicand using perfect fourth power factors.
4√(a2)4
Simplify.
a2
Simplify:
- √a8a6
- 4√x7x3
- 4√y17y5
- Answer
-
- |a|
- |x|
- y3
Simplify:
- √x14x10
- 3√m13m7
- 5√n12n2
- Answer
-
- x2
- m2
- n2
Remember the Quotient to a Power Property? It said we could raise a fraction to a power by raising the numerator and denominator to the power separately.
(ab)m=ambm,b≠0
Quotient Property of Radical Expressions
If n√a and n√b are real numbers, b≠0, and for any integer n≥2 then,
n√ab=n√an√b and n√an√b=n√ab
Simplify: √27m3196
Solution:
Step 1: Simplify the fraction in the radicand, if possible.
27m3196 cannot be simplified.
√27m3196
Step 2: Use the Quotient Property to rewrite the radical as the quotient of two radicals.
We rewrite √27m3196 as the quotient of √27m3 and √196.
√27m3√196
Step 3: Simplify the radicals in the numerator and the denominator.
9m2 and 196 are perfect squares.
√9m2⋅√3m√196
3m√3m14
Simplify: √24p349.
- Answer
-
2|p|√6p7
Simplify: √48x5100.
- Answer
-
2x2√3x5
Simplify a Square Root Using the Quotient Property
- Simplify the fraction in the radicand, if possible.
- Use the Quotient Property to rewrite the radical as the quotient of two radicals.
- Simplify the radicals in the numerator and the denominator.
Simplify:
- √45x5y4
- 3√24x7y3
- 4√48x10y8
Solution:
a.
√45x5y4
We cannot simplify the fraction in the radicand. Rewrite using the Quotient Property.
√45x5√y4
Simplify the radicals in the numerator and the denominator.
√9x4⋅√5xy2
Simplify.
3x2√5xy2
b.
3√24x7y3
The fraction in the radicand cannot be simplified. Use the Quotient Property to write as two radicals.
3√24x73√y3
Rewrite each radicand as a product using perfect cube factors.
3√8x6⋅3x3√y3
Rewrite the numerator as the product of two radicals.
3√(2x2)3⋅3√3x3√y3
Simplify.
2x23√3xy
c.
4√48x10y8
The fraction in the radicand cannot be simplified.
4√48x104√y8
Use the Quotient Property to write as two radicals. Rewrite each radicand as a product using perfect fourth power factors.
4√16x8⋅3x24√y8
Rewrite the numerator as the product of two radicals.
4√(2x2)4⋅4√3x24√(y2)4
Simplify.
2x24√3x2y2
Simplify:
- √80m3n6
- 3√108c10d6
- 4√80x10y4
- Answer
-
- 4|m|√5m|n3|
- 3c33√4cd2
- 2x24√5x2|y|
Simplify:
- √54u7v8
- 3√40r3s6
- 4√162m14n12
- Answer
-
- 3u3√6uv4
- 2r3√5s2
- 3|m3|4√2m2|n3|
Be sure to simplify the fraction in the radicand first, if possible.
Simplify:
- √18p5q732pq2
- 3√16x5y754x2y2
- 4√5a8b680a3b2
Solution:
a.
√18p5q732pq2
Simplify the fraction in the radicand, if possible.
√9p4q516
Rewrite using the Quotient Property.
√9p4q5√16
Simplify the radicals in the numerator and the denominator.
√9p4q4⋅√q4
Simplify.
3p2q2√q4
b.
3√16x5y754x2y2
Simplify the fraction in the radicand, if possible.
3√8x3y527
Rewrite using the Quotient Property.
3√8x3y53√27
Simplify the radicals in the numerator and the denominator.
3√8x3y3⋅3√y23√27
Simplify.
2xy3√y23
c.
4√5a8b680a3b2
Simplify the fraction in the radicand, if possible.
4√a5b416
Rewrite using the Quotient Property.
4√a5b44√16
Simplify the radicals in the numerator and the denominator.
4√a4b4⋅4√a4√16
Simplify.
|ab|4√a2
Simplify:
- √50x5y372x4y
- 3√16x5y754x2y2
- 4√5a8b680a3b2
- Answer
-
- 5|y|√x6
- 2xy3√y23
- |ab|4√a2
Simplify:
- √48m7n2100m5n8
- 3√54x7y5250x2y2
- 4√32a9b7162a3b3
- Answer
-
- 2|m|√35|n3|
- 3xy3√x25
- 2|ab|4√a23
In the next example, there is nothing to simplify in the denominators. Since the index on the radicals is the same, we can use the Quotient Property again, to combine them into one radical. We will then look to see if we can simplify the expression.
Simplify:
- √48a7√3a
- 3√−1083√2
- 4√96x74√3x2
Solution:
a.
√48a7√3a
The denominator cannot be simplified, so use the Quotient Property to write as one radical.
√48a73a
Simplify the fraction under the radical.
√16a6
Simplify.
4|a3|
b.
3√−1083√2
The denominator cannot be simplified, so use the Quotient Property to write as one radical.
3√−1082
Simplify the fraction under the radical.
3√−54
Rewrite the radicand as a product using perfect cube factors.
3√(−3)3⋅2
Rewrite the radical as the product of two radicals.
3√(−3)3⋅3√2
Simplify.
−33√2
c.
4√96x74√3x2
The denominator cannot be simplified, so use the Quotient Property to write as one radical.
4√96x73x2
Simplify the fraction under the radical.
4√32x5
Rewrite the radicand as a product using perfect fourth power factors.
4√16x4⋅4√2x
Rewrite the radical as the product of two radicals.
4√(2x)4⋅4√2x
Simplify.
2|x|4√2x
Simplify:
- √98z5√2z
- 3√−5003√2
- 4√486m114√3m5
- Answer
-
- 7z2
- −53√2
- 3|m|4√2m2
Simplify:
- √128m9√2m
- 3√−1923√3
- 4√324n74√2n3
- Answer
-
- 8m4
- −4
- 3|n|4√2
Access these online resources for additional instruction and practice with simplifying radical expressions.
- Simplifying Square Root and Cube Root with Variables
- Express a Radical in Simplified Form-Square and Cube Roots with Variables and Exponents
- Simplifying Cube Roots
Key Concepts
- Simplified Radical Expression
- For real numbers a,m and n≥2
n√a is considered simplified if a has no factors of mn
- For real numbers a,m and n≥2
- Product Property of nth Roots
- For any real numbers, n√a and n√b, and for any integer n≥2
n√ab=n√a⋅n√b and n√a⋅n√b=n√ab
- For any real numbers, n√a and n√b, and for any integer n≥2
- How to simplify a radical expression using the Product Property
- Find the largest factor in the radicand that is a perfect power of the index.
Rewrite the radicand as a product of two factors, using that factor. - Use the product rule to rewrite the radical as the product of two radicals.
- Simplify the root of the perfect power.
- Find the largest factor in the radicand that is a perfect power of the index.
- Quotient Property of Radical Expressions
- If n√a and n√b are real numbers, b≠0, and for any integer n≥2 then, n√ab=n√an√b and n√an√b=n√ab
- How to simplify a radical expression using the Quotient Property.
- Simplify the fraction in the radicand, if possible.
- Use the Quotient Property to rewrite the radical as the quotient of two radicals.
- Simplify the radicals in the numerator and the denominator.