Question

Which functions are equivalent to f(x)=^4to the square root of 162^x

Answer 1

The function equivalent to
`\(f(x) = \sqrt[4]{162}\)` is
`\(f(x) = [3 * (2^{(1)/(4)})]^x\)`, represented by option E.

Let's simplify the original function
`\(f(x) = \sqrt[4]{162}\)` first.

`\[ f(x) = \sqrt[4]{162} \]`

Since
`\(162 = 2 * 3^4\)`, we can rewrite it as:

`\[ f(x) = \sqrt[4]{2 * 3^4} \]`

Now, using the property
`\(a^(mn) = (a^m)^n\)`, we can simplify it further:

`\[ f(x) = (2 * 3^4)^{(1)/(4)} \]\[ f(x) = (2^{(1)/(4)} * 3)^{(1)/(4)} \]`

This simplifies to:

`\[ f(x) = 3^{(1)/(4)} * 2^{(1)/(16)} \]`

Now, let's compare this to the given options:

A)
`\(f(x) = (162)/(4^x)\)` - Not equivalent.

B)
`\(f(x) = (3 * 4^{(√(2))/(x)})\)` - Not equivalent.

C)
`\(f(x) = 9 * (4^(√(2) * x))\)` - Not equivalent.

D)
`\(f(x) = (162)/(4^x)\)` - Not equivalent.

E)
`\(f(x) = [3 * (2^{(1)/(4)})]^x\)` - Equivalent.

Therefore, the function equivalent to
`\(f(x) = \sqrt[4]{162}\) is \(f(x) = [3 * (2^{(1)/(4)})]^x\)`, so the correct answer is option E.

The complete question is:
Which functions are equivalent to f(x)= 4√162? Check all that apply

A)f(x)= 162 x/4

B)f(x)=(3 4√2)x

C)f(x)=9 4√2 x

D)f(x)=162 4/x

E)f(x)=[3 (2 1/4)]x