How To Simplify Radicals

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A radical of a positive real quantity is called a surd if its value cannot exactly determine. Radical is represented by root √, whether it can be square root or cube root. Thus, each of the quantities √3, (16) ^ (2/3), (7) ^ (1/5) etc., a radical.
From the definition it is evident that a radical is an incommensurable quantity, although its value can be determined to any degree of accuracy should not be noted that quantities √4, (27) ^ (1/3), (16 / 81) ^ (1/4) etc., expressed in the form radical are commensurable quantities are not radical because

·         √4 = 2

·         (27) ^ (1/3) = 3
·         (16 / 81) ^ (1/4) = 2 / 3
 
In fact, any root of an algebraic expression regarded as a radical.


 Question 1: - √32 – 2 √18 + 5 √2 + 2 ^ (3/2)
 

Solution: -
√32 – 2 √18 + 5 √2 + 2 ^ (3/2)= √(16 * 2) – 2 √(9*2) + 5 √2 +√(2^3)

                                                =4√2 - 6√2 +5√2+2√2

                                                =(4-6+5+2) √2

                                                =5√2
Answer: - 5√2
 


Question 2: - Simplify: 3√2 x 5 (4)^(1/3) x 4 (8)^(1/4)
 

Solution: -
3√2 x 5 (4) ^(1/3) x 4 (8)^(1/4)    = (3x5x4)x[2^(1/2) x 4^(1/3) x 8^(1/4)]

                                                   = 60 x 2^ (1/2) x 2^(2/3) x 2^(3/4)

                                                   = 60 x 2^ (1/2 + 2/3 + ¾)

                                                   = 60 x 2^ (23/12)

                                                   = 60 x (2^23) ^(1/12)

                                                   =60 x [2^ (12+11)] ^(1/12)

                                                   = 60 x (2^12 x 2^11)^(1/12)

                                                   = 60 x 2 x(2^11)^(1/12)

                                                   = 120 x (2048)^ (1/2)

                                                   = 120 (2048)^(1/2)




 

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