Perform the addition and simplify your answer. 25 3 x 3 + 3 x 3 \sqrt { \frac { 25 } { 3 x ^ { 3 } } } + \sqrt { 3 x ^ { 3 } } 3 x 3 25 + 3 x 3 A) ( 5 + 3 x 3 ) 3 x 3 x 2 \frac { \left( 5 + 3 x ^ { 3 } \right) \sqrt { 3 x } } { 3 x ^ { 2 } } 3 x 2 ( 5 + 3 x 3 ) 3 x B) ( 5 + 3 x 4 ) 3 x 3 x 3 \frac { \left( 5 + 3 x ^ { 4 } \right) \sqrt { 3 x } } { 3 x ^ { 3 } } 3 x 3 ( 5 + 3 x 4 ) 3 x C) ( 5 + 3 x 2 ) 3 3 x \frac { \left( 5 + 3 x ^ { 2 } \right) 3 } { 3 x } 3 x ( 5 + 3 x 2 ) 3 D) ( 5 + 3 x 3 ) 3 x 5 x 2 \frac { \left( 5 + 3 x ^ { 3 } \right) \sqrt { 3 x } } { 5 x ^ { 2 } } 5 x 2 ( 5 + 3 x 3 ) 3 x E) ( 3 + 5 x 2 ) 3 5 x \frac { \left( 3 + 5 x ^ { 2 } \right) \sqrt { 3 } } { 5 x } 5 x ( 3 + 5 x 2 ) 3

Question

Perform the addition and simplify your answer.   25 3 x 3 + 3 x 3 \sqrt { \frac { 25 } { 3 x ^ { 3 } } } + \sqrt { 3 x ^ { 3 } } 3 x 3 25 ​ ​ + 3 x 3 ​ A)    ( 5 + 3 x 3 )  3 x 3 x 2 \frac { \left( 5 + 3 x ^ { 3 } \right)  \sqrt { 3 x } } { 3 x ^ { 2 } } 3 x 2 ( 5 + 3 x 3 )  3 x ​ ​ B)    ( 5 + 3 x 4 )  3 x 3 x 3 \frac { \left( 5 + 3 x ^ { 4 } \right)  \sqrt { 3 x } } { 3 x ^ { 3 } } 3 x 3 ( 5 + 3 x 4 )  3 x ​ ​ C)    ( 5 + 3 x 2 )  3 3 x \frac { \left( 5 + 3 x ^ { 2 } \right)  3 } { 3 x } 3 x ( 5 + 3 x 2 )  3 ​ D)    ( 5 + 3 x 3 )  3 x 5 x 2 \frac { \left( 5 + 3 x ^ { 3 } \right)  \sqrt { 3 x } } { 5 x ^ { 2 } } 5 x 2 ( 5 + 3 x 3 )  3 x ​ ​ E)    ( 3 + 5 x 2 )  3 5 x \frac { \left( 3 + 5 x ^ { 2 } \right)  \sqrt { 3 } } { 5 x } 5 x ( 3 + 5 x 2 )  3 ​ ​

Carla Tolbert · Accepted Answer

Final Answer : A, Explanation :   To simplify the expression, first express both terms under a common radical and denominator. The first term is   25 3 x 3 = 5 3 x 3 \sqrt{\frac{25}{3x^3}} = \frac{5}{\sqrt{3x^3}} 3 x 3 25 ​ ​ = 3 x 3 ​ 5 ​  , which can be rewritten as   5 3 x 3 x 3 / 2 \frac{5\sqrt{3x}}{3x^{3/2}} 3 x 3/2 5 3 x ​ ​  . The second term is   3 x 3 = 3 x 3 3 x 3 x = 3 x 3 / 2 3 x 3 x 3 / 2 \sqrt{3x^3} = \frac{\sqrt{3x^3}\sqrt{3x}}{\sqrt{3x}} = \frac{3x^{3/2}\sqrt{3x}}{3x^{3/2}} 3 x 3 ​ = 3 x ​ 3 x 3 ​ 3 x ​ ​ = 3 x 3/2 3 x 3/2 3 x ​ ​  . Adding these under a common denominator of   3 x 3 / 2 3x^{3/2} 3 x 3/2   gives   5 3 x + 3 x 3 / 2 3 x 3 x 3 / 2 \frac{5\sqrt{3x} + 3x^{3/2}\sqrt{3x}}{3x^{3/2}} 3 x 3/2 5 3 x ​ + 3 x 3/2 3 x ​ ​  , which simplifies to   ( 5 + 3 x 3 ) 3 x 3 x 3 / 2 \frac{(5 + 3x^3)\sqrt{3x}}{3x^{3/2}} 3 x 3/2 ( 5 + 3 x 3 ) 3 x ​ ​  , and since   x 3 / 2 = x 1.5 = x 1 + 0.5 = x 1 x 0.5 = x x x^{3/2} = x^{1.5} = x^{1+0.5} = x^1x^{0.5} = x\sqrt{x} x 3/2 = x 1.5 = x 1 + 0.5 = x 1 x 0.5 = x x ​  , the denominator simplifies to   3 x 2 3x^2 3 x 2  , making the correct answer   ( 5 + 3 x 3 ) 3 x 3 x 2 \frac{(5 + 3x^3)\sqrt{3x}}{3x^2} 3 x 2 ( 5 + 3 x 3 ) 3 x ​ ​  .

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