Question

use the properties of exponents to write the function in the form f(t)=ka^t where k is a constant.(1/3)^(2-3t)

Answer 1

We are given the following function

`((1)/(3))^(2-3t)`

Let us re-write this function in the following form.

`f(t)=ka^t`

Where k is a constant.

Step 1:

Split the powers using the multiplication rule of exponents.

`a^(x+y)=a^x\cdot a^y`

Applying the above rule, the function becomes

`((1)/(3))^(2-3t)=((1)/(3))^2\cdot((1)/(3))^(-3t)`

Further simplifying, the function becomes

`((1)/(3))^2\cdot((1)/(3))^(-3t)=(1)/(9)\cdot((1)/(3))^(-3t)`

Step 2:

Apply the power rule of exponents

`a^(xy)=(a^x)^y`

So, the function becomes

`(1)/(9)\cdot((1)/(3))^(-3t)=(1)/(9)\cdot(((1)/(3))^(-3))^t`

Further simplifying the function becomes

`(1)/(9)\cdot(((1)/(3))^(-3))^t=(1)/(9)\cdot(27^{})^t`

Therefore, the function is

`f(t)=(1)/(9)\cdot27^t`

Where k = 1/9 and a = 27