To solve the expression [tex]\( 256^{1/4} \)[/tex], let's break it down step-by-step.
1. Identify the base and the exponent:
- The base is [tex]\( 256 \)[/tex].
- The exponent is [tex]\( \frac{1}{4} \)[/tex], which means we need to find the fourth root of 256.
2. Understanding the fourth root:
- The fourth root of a number [tex]\( x \)[/tex] is a number [tex]\( y \)[/tex] such that [tex]\( y^4 = x \)[/tex].
- So, we need to find a number [tex]\( y \)[/tex] such that [tex]\( y^4 = 256 \)[/tex].
3. Checking the possible answers:
- Let’s evaluate each option to see which one, when raised to the power of 4, equals 256.
- A. [tex]\( 16^4 \)[/tex]:
[tex]\[
16^4 = 16 \times 16 \times 16 \times 16 = 256 \times 256 = 65536
\][/tex]
This value is much larger than 256.
- B. [tex]\( 8^4 \)[/tex]:
[tex]\[
8^4 = 8 \times 8 \times 8 \times 8 = 64 \times 64 = 4096
\][/tex]
This value is also much larger than 256.
- C. [tex]\( 4^4 \)[/tex]:
[tex]\[
4^4 = 4 \times 4 \times 4 \times 4 = 16 \times 16 = 256
\][/tex]
This value matches 256.
- D. [tex]\( 64^4 \)[/tex]:
[tex]\[
64^4 = 64 \times 64 \times 64 \times 64
\][/tex]
This value will be much larger than 256 as 64 is a large number.
4. Conclusion:
- The correct value is [tex]\( 4 \)[/tex] because [tex]\( 4^4 = 256 \)[/tex].
Thus, the value of the expression [tex]\( 256^{1/4} \)[/tex] is [tex]\( 4 \)[/tex].
Therefore, the correct option is:
C. 4.
