Let's simplify each expression shown in the exercise:
Option a
Given:
[tex]x^3\cdot x^{-3}[/tex]
You can apply the Product of powers property. This states the following:
[tex]b^n\cdot b^m=b^{(n+m)}[/tex]
Where "b" is the base and "n" and "m" are exponents.
Then, you get:
[tex]x^3\cdot x^{-3}=x^{(3+(-3))}=x^{(3-3)}=x^0[/tex]
By definition:
[tex]b^0=1[/tex]
Therefore:
[tex]x^0=1[/tex]
The expression given in Option a is equal to 1.
Option b
Knowing that any number or expression with exponent zero is equal to 1, you get that:
[tex]1001^0=1[/tex]
The expression given in Option b is equal to 1.
Option c
Given:
[tex]\frac{a^2b}{ba^2}[/tex]
You can notice that the numerator and the denominator are equal, then:
[tex]\frac{a^2b}{ba^2}=1[/tex]
The expression given in Option c is equal to 1.
Option d
Given:
[tex]\frac{y^2}{y^{-2}}[/tex]
You can simplify it using the Quotient of powers property, which states that:
[tex]\frac{b^m}{b^n}=b^{(m-n)}[/tex]
Then, you get:
[tex]y^{(2-(-2))}=y^{(2+2)}=y^4[/tex]
The expression given in Option d is not equal to 1.
The answer is: Option d.
