To find the difference between [tex]\(126^{1/4}\)[/tex] and [tex]\(78^{2/3}\)[/tex], we follow these steps:
1. Evaluate [tex]\(126^{1/4}\)[/tex]:
- Calculate the fourth root of 126.
- This gives us approximately [tex]\(3.350368958834508\)[/tex].
2. Evaluate [tex]\(78^{2/3}\)[/tex]:
- Calculate [tex]\(78\)[/tex] raised to the power of [tex]\(\frac{2}{3}\)[/tex], which involves finding the cubic root of [tex]\(78\)[/tex] and then squaring the result.
- This gives us approximately [tex]\(18.25561221028858\)[/tex].
3. Calculate the difference:
- Subtract the value of [tex]\( 78^{2/3} \)[/tex] from [tex]\( 126^{1/4} \)[/tex]:
[tex]\[
3.350368958834508 - 18.25561221028858 = -14.90524325145407
\][/tex]
Taking into consideration the provided numerical values:
[tex]\[
126^{1/4} \approx 3.350368958834508
\][/tex]
and
[tex]\[
78^{2/3} \approx 18.25561221028858
\][/tex]
the difference between these values is:
[tex]\[
3.350368958834508 - 18.25561221028858 = -14.90524325145407.
\][/tex]
None of the given answers A), B), C), or D) corresponds to this value. Hence, the correct choice is not among the provided options.
