Answer/Step-by-step explanation:
1. [tex] \frac{(-2)^{-5}}{(-2)^{-10}} [/tex]
Apply the Quotient rule: i.e. [tex] \frac{x^n}{x^m} = x^{n - m} [/tex]
[tex] = (-2)^{-5 - (-10)} = (-2)^5} = -32 [/tex]
2. [tex] 2^{-1} * 2^{-4} [/tex]
Apply the product rule: i.e. [tex] x^n * x^m = x^{n + m} [/tex].
[tex] = 2^{-1 + (-4)} = 2^{-1 - 4} [/tex]
[tex] = 2^{-5} [/tex]
Apply the negative exponent rule: i.e. [tex] x^{-n} = \frac{1}{x^n} [/tex]
[tex] = 2^{-5} = \frac{1}{2^5} [/tex]
[tex] = \frac{1}{32} [/tex]
3. [tex] (-\frac{1}{2})^3 * (-\frac{1}{2})^2 [/tex]
Apply product rule
[tex] = (-\frac{1}{2})^{3 + 2} [/tex]
[tex] = (-\frac{1}{2})^{5} [/tex]
[tex] = -\frac{1^5}{2^5} [/tex]
[tex] = -\frac{1}{32} [/tex]
4. [tex] \frac{2}{2^{-4}} [/tex]
Apply the rules of 1 and quotient rule
[tex] = 2^{1 - (-4)} [/tex]
[tex] = 2^{1 + 4} [/tex]
[tex] = 2^{5} = 32 [/tex]
