What is the simplified base of the function f(x) = One-fourth (Root Index 3 StartRoot 108 EndRoot) Superscript x? 3 3RootIndex 3 StartRoot 4 EndRoot 6RootIndex 3 StartRoot 3 End…

Question

asked Aug 5, 2020 112k views

2 Answers

Andreas Wolf · Answer 1 · 2020-08-10T22:57:34+0000

Answer:

Simplified base would be 3∛4

Explanation:

Given exponential function,

$f(x)=(1)/(4)(\sqrt[3]{108})^x$

Since, in an exponential function
f(x)=ab^x

b is called base.

∵ 108 = 2 × 2 × 3 × 3 × 3,

Or 108 = 4 × 3³,

$\implies \sqrt[3]{108} = \sqrt[3]{4* 3^3}$

$\sqrt[3]{108}=\sqrt[3]{4}* \sqrt[3](3^3)$ ( Using
$\sqrt[n]{ab}=\sqrt[n]{a}* \sqrt[n]{b}$ )

$\sqrt[3]{108}=\sqrt[3]{4}* (3^3)^(1)/(3)$

$\sqrt[3]{108}=\sqrt[3]{4}* 3^{3* (1)/(3)}$ ( Using power of power property of exponent )

$\sqrt[3]{108}=3\sqrt[3]{4}$

Hence, the required simplified base would be 3∛4

i.e. SECOND option is correct.