Question

Write each expression as a^n where a ∈ R and n ∈ Z (there are 2 images)

Answer 1

16c. 3^-10 or 9^-5

16d. 3^7

Explanation:

You want to simplify two exponential expressions, putting them in the form a^b with real base and integer exponent.

Rules of exponents

The relevant rules of exponents are ...

(a^b)(a^c) = a^(b+c)

(a^b)^c = a^(bc)

(ab)^c = (a^c)(b^c)

16c

`9^(-6)\cdot3^4\cdot(9^3\cdot3^(-4))^(-1)=9^(-6)\cdot3^4\cdot9^(-3)\cdot3^4\\\\=9^(-6-3)\cdot3^(4+4)=(3^2)^(-9)\cdot3^8=3^(-18+8)=\boxed{3^(-10)}\text{ or }\boxed{9^(-5)}`

16d

`(((1)/(3))^(-3)\cdot3^(-2)\cdot8)/(2^3\cdot27)\cdot(3^3)^3=((3^(-1))^(-3)\cdot3^(-2)\cdot8)/(3^3\cdot8)\cdot3^9=(3^3\cdot3^(-2)\cdot8)/(3^3\cdot8)\cdot3^9\\\\=3^(-2+9)=\boxed{3^(7)}`

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