properties of radicals

Question

Using properties of radicals

Simplify the expression.

1)(5)^1/3×(25)^1/3

2)(3)^1/4×(27)^1/4

3)(64)^1/2×(4)^1/2

Answer 1

1) The expression is (5)^1/3×(25)^1/3

=(5)^1/3×(5²)^1/3                                     (5²=25)

The property(a^m)ⁿ =a^mn

=(5)^1/3×(5)^2/3                            ((5²)^1/3 = (5)^2/3 )

The property a^m×aⁿ=a^m+n

=(5)^(1/3)+(2/3)                                ((1/3)+(2/3)=3/3=1)

=5¹                                            

=5

(5)^1/3×(25)^1/3 =5

2) (3)^1/4×(27)^1/4

The expression is (3)^1/4×(27)^1/4

=(3)^1/4×(27)^1/4                           (3³=27)

The property(a^m)ⁿ =a^mn

=(3)^1/4×(3)^3/4                             (  (3³)^1/4 = (3)^3/4 )

The property a^m×aⁿ=a^m+n

=(3)^(1/4)+(3/4)                                ((1/4)+(3/4)=4/4=1)

= 3¹

=3

(3)^1/4×(27)^1/4 = 3.

3) (64)^1/2×(4)^1/2

The expression is  (64)^1/2×(4)^1/2  

=(4³)^1/2×(4)^1/2

The property(a^m)ⁿ =a^mn

=(4)^3/2   Ã—(4)^1/2

The property a^m×aⁿ=a^m+n

=(4)^3/2+(1/2)

=(4)²                            

=16

(64)^1/2×(4)^1/2 = 16.