A negative coefficient of a term with a rational exponent can mean that we either (1) apply the rational exponent and then take the opposite of the result, or (2) the rational exponent applies to a negative term. In case 2 of rational exponents with negative coefficients, the answer will be not real if the denominator of the exponent is even. If the root is odd, the answer will be a negative number.
What we're going to talk about now is negative coefficients with radicals okay? Now let's start calling radicals by they having the same meaning as with a fractional exponents because really they're one and the same thing.
Okay, so what I have is just a number of examples on the board and we're going to talk about basically this sign the the positive negative aspects of most of examples, so 64 to the one half, remember one half is just the same thing as the square root, so really is the same as the square root of 64 and that's 8. Okay, -64 to the one half, so really the only difference between this and the one above it is the negative sign. Remember that order of operations tells us do our exponents before our plus or minus or even multiplication so really we still have to do 64 to the one half first square root of 64 which is 8 and then the negative comes out on front so this leaves us with a negative 8.
Okay, negative 64 to the one half. Now the negative is actually associated with this one half so now we're taking the square root of negative 64. We can't take the square root of a negative number so this is going to be not real, okay? This last example is exactly the same as the one above but with this negative sign out on the front so again the negative 64 to the one half this is the square root of a negative number which is not real so this is not real as well okay? So basically making sure we know when this negative is associated with that power, with that square root, with that rational exponent.
Okay, let's take a look at some odd roots okay? We're talking about one over 2 back there that means we're taking the square roots. One over 3 means we're dealing with cube roots okay? The difference with cube roots is we can actually have the cube root of a negative number. 27 to the one third cube root of 27 this is just going to be 3, -27 to the one third again the one third is not associated with the negative sign order of operations tell us we have to do the exponent first, 27 to one third is 3 negative on the front tells us this is negative 3, negative 27 to the one third we're now dealing with the cube root of negative 27 cube root negative of 27, we can take the cube root of a negative number which is going to be -3. Our last one the inside is exactly the same as what we were just above so this inside is going to be -3 but we have a negative out on front so we have negative negative 3, two negatives cancel just giving us 3.
So what we really have is just a bunch of examples on how we consider the negative sign making sure we know when its associated with the power with the root and when its just hanging out on the outside.