INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith

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For more information, please see full course syllabus of Algebra 1

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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}
• √{176}
• √{4 ×44}
• √{4 ×4 ×11}
• √{4 ×4} ×√{11}
• 2 ×2 ×√{11}
4√{11}
Simplify:
√7 ×√{40}
• √{12 ×40}
• √{280}
• √{4 ×70}
• √4 ×√{70}
2√{70}
Simplify:
√{12} ×√{40}
• √{12 ×40}
• √{480}
• √{4 ×120}
• √{4 ×4 ×30}
• √{4 ×4} ×√{30}
• 2 ×2 ×√{30}
4√{30}
Simplify:
√{14x3y2} ×√{6x2y2}
• √{14x3y2 ×6x2y2}
• √{84x5y4}
• √{4 ×21x4xy4}
• √{4x4y4} ×√{21x}
• 2Px2Py2P ×√{21x}
2x2y2 ×√{21x}
Simplify:
√{6a2b3c4} ×√{20a2b3c}
• √{6a2b3c4 ×20a2b3c}
• √{120a4b6c5}
• √{4 ×30a4b6c4c}
• √{4a4b6c4} ×√{30c}
• 2| a2 || b3 || c2 | ×√{30c}
2a2| b3 |c2√{30c}
Simplify:
√{15g3hi2} ×√{10g2h4i3}
• √{15g3hi2 ×10g2h4i3}
• √{150g5h5i5}
• √{25 ×6g4gh4hi4i}
• √{25g4h4i4} ×√{6ghi}
• 5| g2 || h2 || i2 | ×√{6ghi}
5g2h2i2√{6ghi}
Simplify:
√{[10/2]}
• [(√{10} )/(√2 )] ×[(√2 )/(√2 )]
• [(√{10 ×2} )/(( √2 )2)]
• [(√{2 ×2 ×5} )/2]
[(2√5 )/2]
Simplify:
• [(√{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 )] ×[(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.

### 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