To determine which expression is not equivalent to sqrt(4n^2), we will rewrite it using the Properties of Exponents. Recall that a radical expression can be written as a power with a rational exponent, where the index of the radical is the denominator of the exponent.
sqrt(4n^2) ⇔ (4n^2)^(1 4)
The above is the expression given in choice A. Let's continue simplifying. To do so we will rewrite the exponent as a product of two factors. Then we will use the Power of a Power Property.
We do not know whether n is positive or negative. Therefore, to simplify the obtained expression we must use the absolute value symbol.
( 2sqrt(n^2) ) ^() 12
sqrt(a^2)=|a|
( 2|n| ) ^() 12
Note that the above is the expression given in choice C. Finally, recall that raising an expression to the power of 12 is the same as calculating the square root of the expression.
( 2|n| ) ^() 12 ⇔ sqrt(2|n|)
We obtained the expression given in choice D. Therefore, the expression that is not equivalent to sqrt(4n^2) is the one given in choice B.
