is the function below exponential?

Question

is the function below exponential?

K(x) = 3^x/ √(3·6^x)

write the function in the form K(x)= ab^x and enter the values for a and b 

Answer 1

3^x ⁄ √(3·6^x)

= 3^x ⁄√3 √(6^x)

= 1⁄√3 · 3^x/√(6^x)

= 1⁄√3 · √(3^2x)/√(6x)

= 1⁄√3 · √(3^2x)/√(6^x)

= 1⁄√3 · √(3^2x/6^x)

= 1⁄√3 · √(3^2x/(3·2)^x)

= 1⁄√3 · √(3^2x/(3^x·2^x))

= 1⁄√3 · √(3^x/2^x)

= 1⁄√3 · √[(3/2)^x] = 1⁄√3 · ((3/2)^x)^0.5 = 1⁄√3 · ((3/2)^0.5)^x

= 1⁄√3 · (√(3/2))^x

 

so a = 1⁄√3 = √(3)/3

and b = √(3/2)

 

It's tough handling complex square roots with this text formatter. To keep it cleaner, that's why I used a fractional exponent in the next to last step.