what is the difference between -16^1/4 and (-16)^1/4
-16^1/4 = (-1)(16^1/4) = -2
(-16)^1/4 = [(-1)(16)]^1/4 = (-1)^1/4 (16^1/4) = 2 * (-1)^1/4
not the same at all.
THANK YOU!
To understand the difference between -16^1/4 and (-16)^1/4, let's break down the expressions:
-16^1/4: In this expression, the negative sign applies only to the number 16 and not the exponent. The exponent 1/4 indicates the fourth root of a number. Therefore, -16^1/4 means calculating the fourth root of 16 first and then applying the negative sign.
(-16)^1/4: In this expression, the negative sign is inside the parentheses, which means it applies to the entire expression within the parentheses. The exponent 1/4 still indicates the fourth root of a number. Therefore, (-16)^1/4 means calculating the fourth root of negative 16.
Let's evaluate each expression separately:
-16^1/4:
To calculate the fourth root of 16, we raise 16 to the power of 1 divided by 4.
16^1/4 = √√16 = √2 = ±2
Since the negative sign is applied after the exponentiation, -16^1/4 becomes -(±2), resulting in -2 or +2.
(-16)^1/4:
To calculate the fourth root of -16, we raise -16 to the power of 1 divided by 4.
(-16)^1/4 = √√(-16)
Here comes an important distinction: In this case, taking the fourth root of a negative number does not yield a real number solution. It results in a complex number. Therefore, (-16)^1/4 is not a real number.
In conclusion, the difference between -16^1/4 and (-16)^1/4 is that -16^1/4 can be either -2 or +2, while (-16)^1/4 does not give a real number solution.