Answer :
To determine which of the given expressions is equivalent to [tex]\(\left(\frac{-11}{14}\right)^5\)[/tex], we need to carefully evaluate each option and see how they compare to the original expression.
Given expression:
[tex]\[ \left(\frac{-11}{14}\right)^5 \][/tex]
Let's evaluate each option one by one to see which one matches.
Option A: [tex]\(5 \cdot \frac{-11}{14}\)[/tex]
This expression simplifies to:
[tex]\[ 5 \cdot \frac{-11}{14} = \frac{5 \cdot (-11)}{14} = \frac{-55}{14} \][/tex]
which is clearly not equivalent to [tex]\(\left(\frac{-11}{14}\right)^5\)[/tex].
Option B: [tex]\(\frac{(-11)^5}{14^5}\)[/tex]
This expression can be expanded as:
[tex]\[ \frac{(-11)^5}{14^5} \][/tex]
which is exactly the same as:
[tex]\[ \left(\frac{-11}{14}\right)^5 = \frac{(-11)^5}{14^5} \][/tex]
Option C: [tex]\(\frac{(-11)^5}{14}\)[/tex]
This option simplifies to:
[tex]\[ \frac{(-11)^5}{14} \][/tex]
This is not equivalent to [tex]\(\left(\frac{-11}{14}\right)^5\)[/tex], because in the original expression both the numerator and the denominator are raised to the power of 5, not just the numerator.
Option D: [tex]\(\frac{-11}{14^5}\)[/tex]
This expression simplifies to:
[tex]\[ \frac{-11}{14^5} \][/tex]
which is clearly different from [tex]\(\left(\frac{-11}{14}\right)^5\)[/tex], as only the denominator is raised to the power of 5, not both the numerator and the denominator.
Thus, the only expression that is equivalent to [tex]\(\left(\frac{-11}{14}\right)^5\)[/tex] is:
[tex]\[ \boxed{\frac{(-11)^5}{14^5}} \][/tex]
Hence, the correct answer is option B.
Given expression:
[tex]\[ \left(\frac{-11}{14}\right)^5 \][/tex]
Let's evaluate each option one by one to see which one matches.
Option A: [tex]\(5 \cdot \frac{-11}{14}\)[/tex]
This expression simplifies to:
[tex]\[ 5 \cdot \frac{-11}{14} = \frac{5 \cdot (-11)}{14} = \frac{-55}{14} \][/tex]
which is clearly not equivalent to [tex]\(\left(\frac{-11}{14}\right)^5\)[/tex].
Option B: [tex]\(\frac{(-11)^5}{14^5}\)[/tex]
This expression can be expanded as:
[tex]\[ \frac{(-11)^5}{14^5} \][/tex]
which is exactly the same as:
[tex]\[ \left(\frac{-11}{14}\right)^5 = \frac{(-11)^5}{14^5} \][/tex]
Option C: [tex]\(\frac{(-11)^5}{14}\)[/tex]
This option simplifies to:
[tex]\[ \frac{(-11)^5}{14} \][/tex]
This is not equivalent to [tex]\(\left(\frac{-11}{14}\right)^5\)[/tex], because in the original expression both the numerator and the denominator are raised to the power of 5, not just the numerator.
Option D: [tex]\(\frac{-11}{14^5}\)[/tex]
This expression simplifies to:
[tex]\[ \frac{-11}{14^5} \][/tex]
which is clearly different from [tex]\(\left(\frac{-11}{14}\right)^5\)[/tex], as only the denominator is raised to the power of 5, not both the numerator and the denominator.
Thus, the only expression that is equivalent to [tex]\(\left(\frac{-11}{14}\right)^5\)[/tex] is:
[tex]\[ \boxed{\frac{(-11)^5}{14^5}} \][/tex]
Hence, the correct answer is option B.
