Select the expression that is equivalent to (1)/((3x^(2)-5)^(-(2)/(5))) sqrt((3x^(2)-5)^(5)) root(5)((3x^(2)-5)^(2)) (1)/(sqrt((3x^(2)-5)^(5))) (1)/(root(5)((3x^(2)-5)^(2)))

Question

Select the expression that is equivalent to  (1)/((3x^(2)-5)^(-(2)/(5)))  sqrt((3x^(2)-5)^(5))  root(5)((3x^(2)-5)^(2))  (1)/(sqrt((3x^(2)-5)^(5)))  (1)/(root(5)((3x^(2)-5)^(2)))

Bytelearn · Accepted Answer

Simplify Negative Exponent: To find an equivalent expression, we need to simplify the given expression. The negative exponent indicates that the expression is the reciprocal of the base raised to the positive exponent. (1)/((3x^(2)-5)^(-(2)/(5))) = (3x^(2)-5)^(2/5)Identify Simplified Expression: Now we need to identify which of the provided options matches the simplified expression. The expression (3x^(2)-5)^(2/5) can be written as the 5th root of (3x^(2)-5) squared, which is the same as root(5)((3x^(2)-5)^(2)).Eliminate Incorrect Options: The correct equivalent expression is therefore root(5)((3x^(2)-5)^(2)). We can eliminate the other options as they do not match the simplified expression: - sqrt((3x^(2)-5)^(5)) is not equivalent because it represents the square root, not the 5th root. - (1)/(sqrt((3x^(2)-5)^(5))) is not equivalent because it represents the reciprocal of the square root, not the 5th root. - (1)/(root(5)((3x^(2)-5)^(2))) is not equivalent because it represents the reciprocal of the 5th root squared, which is not the same as the 5th root of the square.

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