Let's evaluate the given expressions step-by-step.
### Part (a): [tex]\( -32^{\frac{1}{5}} \)[/tex]
1. We need to find the fifth root of [tex]\(-32\)[/tex].
2. When dealing with roots of negative numbers, it's important to note that if the root's denominator (in the exponent's fraction) is an odd number, the result will be a real number.
3. Since [tex]\(\frac{1}{5}\)[/tex] means taking the fifth root, we are dealing with an odd root of a negative number.
4. The fifth root of [tex]\(32\)[/tex] is [tex]\( 2 \)[/tex] because [tex]\( 2^5 = 32 \)[/tex].
5. Therefore, the fifth root of [tex]\(-32\)[/tex] is [tex]\(-2\)[/tex].
So, [tex]\( -32^{\frac{1}{5}} = -2 \)[/tex].
### Part (b): [tex]\( (-256)^{\frac{1}{4}} \)[/tex]
1. We need to find the fourth root of [tex]\(-256\)[/tex].
2. When dealing with roots of negative numbers, it's important to note that if the root's denominator (in the exponent's fraction) is an even number, the result will not be a real number (it will involve imaginary numbers instead).
3. Since [tex]\(\frac{1}{4}\)[/tex] means taking the fourth root, we are dealing with an even root of a negative number.
4. The fourth root of [tex]\( -256 \)[/tex] does not yield a real number because the fourth root of a negative number is not defined in the set of real numbers.
So, [tex]\( (-256)^{\frac{1}{4}} \text{ is not a real number} \)[/tex].
To summarize:
(a) [tex]\( -32^{\frac{1}{5}} = -2 \)[/tex]
(b) [tex]\( (-256)^{\frac{1}{4}} \)[/tex] is not a real number
