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### Simplifying Radical Expressions

- A
*radical expression*contains a square root. The expression inside the square root is called a*radicand*. - To simplify a radical expression, extract all perfect squares from the radicand.
- Use the product and quotient properties of square roots to help you simplify radical expressions.
- If the exponent of the variable inside the radical is even and the resulting simplified expression has an odd exponent, take the
*absolute value*of the expression for the simplified expression to guarantee that it is nonnegative. - In simplified form, there can be no radicals in the denominator. Removing such radicals is called
*rationalizing*the denominator. - To rationalize a
*monomial*denominator, simply multiply the numerator and denominator by the radical in the denominator. - To rationalize a
*binomial*denominator, multiply the numerator and denominator by the conjugate of the denominator. The*conjugate*is the same as the original binomial but with the sign between the first term and the second term reversed. - To be in
*simplified form*, there must be no perfect squares or fractions in the radicand and there must be no radicals in the denominator.

### Simplifying Radical Expressions

Simplify:

√8 ×√{22}

√8 ×√{22}

- √{8 ×22}
- √{176}
- √{4 ×44}
- √{4 ×4 ×11}
- √{4 ×4} ×√{11}
- 2 ×2 ×√{11}

4√{11}

Simplify:

√7 ×√{40}

√7 ×√{40}

- √{12 ×40}
- √{280}
- √{4 ×70}
- √4 ×√{70}

2√{70}

Simplify:

√{12} ×√{40}

√{12} ×√{40}

- √{12 ×40}
- √{480}
- √{4 ×120}
- √{4 ×4 ×30}
- √{4 ×4} ×√{30}
- 2 ×2 ×√{30}

4√{30}

Simplify:

√{14x

√{14x

^{3}y^{2}} ×√{6x^{2}y^{2}}- √{14x
^{3}y^{2}×6x^{2}y^{2}} - √{84x
^{5}y^{4}} - √{4 ×21x
^{4}xy^{4}} - √{4x
^{4}y^{4}} ×√{21x} - 2Px
^{2}Py^{2}P ×√{21x}

2x

^{2}y^{2}×√{21x}Simplify:

√{6a

√{6a

^{2}b^{3}c^{4}} ×√{20a^{2}b^{3}c}- √{6a
^{2}b^{3}c^{4}×20a^{2}b^{3}c} - √{120a
^{4}b^{6}c^{5}} - √{4 ×30a
^{4}b^{6}c^{4}c} - √{4a
^{4}b^{6}c^{4}} ×√{30c} - 2| a
^{2}|| b^{3}|| c^{2}| ×√{30c}

2a

^{2}| b^{3}|c^{2}√{30c}Simplify:

√{15g

√{15g

^{3}hi^{2}} ×√{10g^{2}h^{4}i^{3}}- √{15g
^{3}hi^{2}×10g^{2}h^{4}i^{3}} - √{150g
^{5}h^{5}i^{5}} - √{25 ×6g
^{4}gh^{4}hi4i} - √{25g
^{4}h^{4}i^{4}} ×√{6ghi} - 5| g
^{2}|| h^{2}|| i^{2}| ×√{6ghi}

5g

^{2}h^{2}i^{2}√{6ghi}Simplify:

√{[10/2]}

√{[10/2]}

- [(√{10} )/(√2 )] ×[(√2 )/(√2 )]
- [(√{10 ×2} )/(( √2 )
^{2})] - [(√{2 ×2 ×5} )/2]

[(2√5 )/2]

Simplify:

√{[24/7]}

√{[24/7]}

- [(√{24} )/(√7 )] ×[(√7 )/(√7 )]
- [(√{24 ×7} )/(( √7 )
^{2})] - [(√{168} )/7]
- [(√{4 ×42} )/7]

[(2√{42} )/7]

Simplify:

[(4√2 )/(2√3 − 5√2 )]

[(4√2 )/(2√3 − 5√2 )]

- [(4√2 )/(2√3 − 5√2 )] ×[(2√3 + 5√2 )/(2√3 + 5√2 )]
- [(8√2 √3 + 20√2 √2 )/(( 2√3 )
^{2}− ( 5√2 )^{2})] - [(8√{2 ×3} + 20( √2 )
^{2})/(( 2√3 )^{2}− ( 5√2 )^{2})] - [(8√6 + 20 ×2)/(4 ×3 − 25 ×2)]
- [(8√6 + 40)/(12 − 50)]

[(8√6 + 40)/( − 38)]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Simplifying Radical Expressions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Radical Expression 0:12
- Example: Radicand Simplest Form
- Example: Not Simplest Form
- Principal Square Root (Positive)
- Product Property 3:40
- Examples
- Square Roots of Variables with Even Powers 7:01
- Eliminate Radical Sign
- Divide Exponent by 2
- Absolute Value of Result
- Examples
- Quotient Rule 14:12
- Example
- Rationalizing Denominators 16:08
- Example
- Conjugates 18:33
- Example
- Simplest Radical Form 20:58
- Three Criteria
- Example 1: Simplify Expression 21:57
- Example 2: Simplify Expression 25:12
- Example 3: Simplify Expression 31:37
- Example 4: Simplify Expression 35:29

2 answers

Last reply by: Juan Collado

Fri Dec 29, 2017 12:40 PM

Post by asiel nasser on January 28, 2016

where is example 4 answer?????????????????

1 answer

Last reply by: Dr Carleen Eaton

Sun Apr 26, 2015 12:33 PM

Post by Andria Sperry on April 18, 2015

Can't 6/33 be simplified to 2/11?

1 answer

Last reply by: Dr Carleen Eaton

Sat Feb 23, 2013 6:51 PM

Post by Rishabh Kasarla on February 21, 2013

At 41:09, why is it 2 instead of 12 in the answer?

0 answers

Post by Victor Castillo on January 27, 2013

I feel stupid now.

1 answer

Last reply by: Dr Carleen Eaton

Sun Jan 27, 2013 1:15 PM

Post by Victor Castillo on January 27, 2013

What happened to the square root signs?

0 answers

Post by Victor Castillo on January 27, 2013

OMG you lost me here

0 answers

Post by Victor Castillo on January 27, 2013

Please clarify I am so lost.

1 answer

Last reply by: Dr Carleen Eaton

Sun Jan 27, 2013 1:14 PM

Post by Victor Castillo on January 27, 2013

At 33:55 what decides wether the 3 goes inside or outside in this case of the square root sign?This whole lesson is really messing with my head.

1 answer

Last reply by: Dr Carleen Eaton

Sun Jan 27, 2013 1:12 PM

Post by Victor Castillo on January 27, 2013

You also added absolute value signs then you removed them.Why add them in the first place?

1 answer

Last reply by: Dr Carleen Eaton

Sun Jan 27, 2013 1:09 PM

Post by Victor Castillo on January 27, 2013

I am so confused the square root sign disappeared at 30:21 and you say you eliminated the radicals but they are still there.Doesn't X3 contain and Y2 contain a radical?Please clarify.

1 answer

Last reply by: Dr Carleen Eaton

Sun Jan 27, 2013 1:07 PM

Post by Victor Castillo on January 27, 2013

At 20:42 what happens to the 5?Why does it seem you skip steps or at least explanations constantly.It is frustrating!

1 answer

Last reply by: Dr Carleen Eaton

Sun Jan 27, 2013 1:03 PM

Post by Victor Castillo on January 27, 2013

Does it have to do with square root?

0 answers

Post by Victor Castillo on January 27, 2013

not itself...

0 answers

Post by Victor Castillo on January 27, 2013

Why divide the exponent by 2 and n ot itself??

1 answer

Last reply by: Dr Carleen Eaton

Mon Apr 16, 2012 10:05 PM

Post by Diane Dobmeier on April 12, 2012

Could you please explain to me how cubic roots work? I'm trying to do my homework, and I'm stuck on problems like "Reduce", for instance, the cubic root of 16 m to the fifth power, n to the fourth power'. I just watched your video, but I don't understand how to simplify roots when there are letters involved. I understand the numbers part, but the letters are throwing me off. Thanks!