Select the expression that is equivalent to (1)/((2x^(2)+2)^((3)/(2))) (1)/(sqrt((2x^(2)+2)^(3))) sqrt((2x^(2)+2)^(3)) root(3)((2x^(2)+2)^(2)) (1)/(root(3)((2x^(2)+2)^(2)))

Question

Select the expression that is equivalent to  (1)/((2x^(2)+2)^((3)/(2)))  (1)/(sqrt((2x^(2)+2)^(3)))  sqrt((2x^(2)+2)^(3))  root(3)((2x^(2)+2)^(2))  (1)/(root(3)((2x^(2)+2)^(2)))

Bytelearn · Accepted Answer

Simplify Expression: We need to simplify the given expression (1)/((2x^(2)+2)^((3)/(2))) to match one of the provided options.Understand Exponent: First, let's understand the exponent (3/2). It means that we have a square root and a cube. The square root is the 1/2 part, and the cube is the 3 part. So, (2x^(2)+2)^((3)/(2)) is the same as taking the square root of (2x^(2)+2) and then cubing the result.Rewrite Expression: The expression (1)/((2x^(2)+2)^((3)/(2))) can be rewritten as (1)/(sqrt((2x^(2)+2)^(3))) because taking the square root and then cubing is the same as raising to the power of 3/2.Compare with Options: Now let's compare the rewritten expression with the provided options. The first option is exactly the same as our rewritten expression: (1)/(sqrt((2x^(2)+2)^(3))).Option 1: The second option, sqrt((2x^(2)+2)^(3)), is not equivalent because it does not have the reciprocal (1/) part.Option 2: The third option, root(3)((2x^(2)+2)^(2)), is not equivalent because it represents the cube root of the quantity (2x^(2)+2) squared, which is not the same as our expression.Option 3: The fourth option, (1)/(root(3)((2x^(2)+2)^(2))), is not equivalent because it represents the reciprocal of the cube root of the quantity (2x^(2)+2) squared, which is different from the cube of the square root in our expression.

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