To solve for the values of [tex]\( \left( 16 \right)^{\frac{1}{4}} \)[/tex], we need to determine the fourth roots of 16. Let's go through this step-by-step:
1. Understanding the Problem:
- We need to find the numbers which, when raised to the fourth power, give the value of 16.
- In mathematical form, we are searching for [tex]\( x \)[/tex] such that [tex]\( x^4 = 16 \)[/tex].
2. Identifying Possible Solutions:
- Let's consider both positive and negative numbers, since raising both a positive or a negative number to an even power will result in a positive number.
3. Positive Solution:
- Consider the number 2:
[tex]\[
2^4 = 2 \times 2 \times 2 \times 2 = 16
\][/tex]
- So, 2 is one solution.
4. Negative Solution:
- Now consider the negative number -2:
[tex]\[
(-2)^4 = (-2) \times (-2) \times (-2) \times (-2) = 16
\][/tex]
- Because multiplying four negative twos results in a positive 16, -2 is also a solution.
5. Listing all Real Solutions:
- We have found that both 2 and -2 satisfy the equation [tex]\( x^4 = 16 \)[/tex]. Therefore, [tex]\( \pm 2 \)[/tex] are the real solutions.
6. Conclusion:
- The values of [tex]\( \left( 16 \right)^{\frac{1}{4}} \)[/tex] are [tex]\( \pm 2 \)[/tex].
Therefore, the correct answer to the question is:
A. [tex]\( \pm 2 \)[/tex]
