Respuesta :
Answer: Choice C
RootIndex 12 StartRoot 8 EndRoot Superscript x
12th root of 8^x = (12th root of 8)^x
[tex]\sqrt[12]{8^{x}} = \left(\sqrt[12]{8}\right)^{x}[/tex]
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Explanation:
The general rule is
[tex]\sqrt[n]{x} = x^{1/n}[/tex]
so any nth root is the same as having a fractional exponent 1/n.
Using that rule we can say the cube root of 8 is equivalent to 8^(1/3)
[tex]\sqrt[3]{8} = 8^{1/3}[/tex]
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Raising this to the power of (1/4)x will have us multiply the exponents of 1/3 and (1/4)x like so
(1/3)*(1/4)x = (1/12)x
In other words,
[tex]\left(8^{1/3}\right)^{(1/4)x} = 8^{(1/3)*(1/4)x}[/tex]
[tex]\left(8^{1/3}\right)^{(1/4)x} = 8^{(1/12)x}[/tex]
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From here, we rewrite the fractional exponent 1/12 as a 12th root. which leads us to this
[tex]8^{(1/12)x} = \sqrt[12]{8^{x}} [/tex]
[tex]8^{(1/12)x} = \left(\sqrt[12]{8}\right)^{x} [/tex]
