Question

What is the simplified base for the function f(x) = 2(3√27(2x)?

2
3
9
18

Answer 1

Option C is correct

9 the simplified base for the given function f(x)

Explanation:

Using exponent rules:

`(x^m)^n = x^(mn)`

`\sqrt[n]{x^b} = x^{(b)/(n)}`

Given the function:

`f(x) = 2\sqrt[3]{27^(2x)}`

We can write 27 as:

`27 = 3 \cdot 3 \cdot 3 = 3^3`

then;

`f(x) = 2\sqrt[3]{(3^3)^(2x)}`

Apply the exponent rules:

`f(x) = 2\sqrt[3]{3^(6x)}`

Apply the exponent rules:

`f(x) =2 \cdot (3^(6x))^{(1)/(3)} = 2 \cdot 3^(2x)`

⇒
`f(x) = 2 \cdot (3^2)^x = 2 \cdot 9^x`

⇒
`f(x) =2 \cdot 9^x`

On comparing with exponential function
`f(x) = ab^x` where, b is base of the exponent function, then

b = 9

Therefore, the simplified base for the given function is, 9

Answer 2

option C is correct i.e. 9

Explanation:

We have given that :
`f(x)=2 \sqrt[3]{27^(2x)}`

To find : The simplified base of the function f(x)

Solution:

Now, we solve the equation

`f(x)=2 \sqrt[3]{27^(2x)}`

`f(x)=2(27^x)^{(2)/(3)}`

`f(x)=2(3^(2x))`

`f(x)=2((3^2)^(x))`

`f(x)=2(9^(x))`

Therefore, the simplified base of the function f(x) is 9