\sqrt[n]{x}

Question

`\sqrt[n]{x}`

Answer 1

C.
`4\log_w(x^2-6)-(1)/(3) \log_w(x^2+8)`

Explanation:

The given logarithmic expression is

`\log_w\frac{(x^2-6)^4}{\sqrt[3]{x^2+8} }`

Recall and use the quotient law of logarithms;

`\log_a((m)/(n) )=\log_a(m )-\log_a(n)`

`=\log_w(x^2-6)^4-\log_w\sqrt[3]{x^2+8}`

`=\log_w(x^2-6)^4-\log_w(x^2+8)^{(1)/(3)}`

Recall and use the power rule of logarithms:
`\log_aM^n=n\log_aM`

`=4\log_w(x^2-6)-(1)/(3) \log_w(x^2+8)`

The correct choice is C.

Answer 2

option C

`log_(w)4(x^(2)-6)}-(1)/(3) ({x^(2)+8)}`

Explanation:

Given in the question an expression

`log_(w)\frac{(x^(2)-6)^4}{\sqrt[3]{x^(2)+8}}`

Step1

Apply logarithm subtraction rule:

`log_(w)(m)/(n)=log_(w)m-n`

`log_(w)(x^(2)-6)^4}-{\sqrt[3]{x^(2)+8}`

Step2

Apply logarithm power rule

`log_(w)x^(n)=nlog_(w)x`

`log_(w)4(x^(2)-6)}-(1)/(3) ({x^(2)+8)}`

as

`\sqrt[n]{x}=(x)^(1)/(n)`