Answer :
Let's analyze and simplify both expressions term by term to determine whether Lola is correct.
### Expression A:
[tex]\[ \left[ \left( \frac{a}{b} \right)^{-4} \right]^0 \][/tex]
1. Inner Expression: [tex]\(\left( \frac{a}{b} \right)^{-4}\)[/tex]
- A fraction raised to a power with a negative exponent means we take the reciprocal and then raise it to the corresponding positive power:
[tex]\[ \left( \frac{a}{b} \right)^{-4} = \left( \frac{b}{a} \right)^4. \][/tex]
2. Outer Expression: [tex]\(\left(\cdots\right)^0\)[/tex]
- Any nonzero number raised to the power of 0 is 1:
[tex]\[ \left[ \left( \frac{b}{a} \right)^4 \right]^0 = 1. \][/tex]
So, the value of Expression A is [tex]\(1\)[/tex].
### Expression B:
[tex]\[ \left[ \left( \frac{a}{b} \right)^0 \right]^{-4} \][/tex]
1. Inner Expression: [tex]\(\left( \frac{a}{b} \right)^0\)[/tex]
- Any number raised to the power of 0 is 1:
[tex]\[ \left( \frac{a}{b} \right)^0 = 1. \][/tex]
2. Outer Expression: [tex]\(1^{-4}\)[/tex]
- 1 raised to any power is still 1:
[tex]\[ \left( 1 \right)^{-4} = 1. \][/tex]
So, the value of Expression B is [tex]\(1\)[/tex].
### Conclusion:
Since both Expression A and Expression B evaluate to [tex]\(1\)[/tex], Lola is correct in stating that both expressions have the same value.
Therefore, the correct statement is:
- Lola is correct because each expression has a value of 1.
### Expression A:
[tex]\[ \left[ \left( \frac{a}{b} \right)^{-4} \right]^0 \][/tex]
1. Inner Expression: [tex]\(\left( \frac{a}{b} \right)^{-4}\)[/tex]
- A fraction raised to a power with a negative exponent means we take the reciprocal and then raise it to the corresponding positive power:
[tex]\[ \left( \frac{a}{b} \right)^{-4} = \left( \frac{b}{a} \right)^4. \][/tex]
2. Outer Expression: [tex]\(\left(\cdots\right)^0\)[/tex]
- Any nonzero number raised to the power of 0 is 1:
[tex]\[ \left[ \left( \frac{b}{a} \right)^4 \right]^0 = 1. \][/tex]
So, the value of Expression A is [tex]\(1\)[/tex].
### Expression B:
[tex]\[ \left[ \left( \frac{a}{b} \right)^0 \right]^{-4} \][/tex]
1. Inner Expression: [tex]\(\left( \frac{a}{b} \right)^0\)[/tex]
- Any number raised to the power of 0 is 1:
[tex]\[ \left( \frac{a}{b} \right)^0 = 1. \][/tex]
2. Outer Expression: [tex]\(1^{-4}\)[/tex]
- 1 raised to any power is still 1:
[tex]\[ \left( 1 \right)^{-4} = 1. \][/tex]
So, the value of Expression B is [tex]\(1\)[/tex].
### Conclusion:
Since both Expression A and Expression B evaluate to [tex]\(1\)[/tex], Lola is correct in stating that both expressions have the same value.
Therefore, the correct statement is:
- Lola is correct because each expression has a value of 1.
