What Is The Simplified Base Of The Function $f(x)=\frac{1}{4}(\sqrt{108})^x$?A. 3 B. $3 \sqrt[3]{4}$ C. $6 \sqrt[3]{3}$ D. 27

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Understanding the Function

The given function is $f(x)=\frac{1}{4}(\sqrt{108})^x$. To simplify the base of this function, we need to start by simplifying the expression inside the parentheses, which is $\sqrt{108}$.

Simplifying the Square Root

The square root of 108 can be simplified by finding the prime factors of 108. We can break down 108 into its prime factors: $108 = 2^2 \times 3^3$.

Simplifying the Square Root of 108

Using the prime factors of 108, we can rewrite the square root of 108 as $\sqrt{108} = \sqrt{2^2 \times 3^3}$. This can be further simplified to $\sqrt{108} = 2 \times \sqrt{3^3}$.

Simplifying the Cube Root

The cube root of $3^3$ is simply 3, since $3^3 = 27$ and the cube root of 27 is 3. Therefore, we can simplify the expression $\sqrt{108}$ to $2 \times 3 = 6$.

Simplifying the Function

Now that we have simplified the expression inside the parentheses, we can rewrite the function as $f(x)=\frac{1}{4}(6)^x$.

Simplifying the Base

To simplify the base of the function, we can rewrite the expression $\frac{1}{4}(6)^x$ as $\frac{1}{4} \times 6^x$.

Simplifying the Fraction

The fraction $\frac{1}{4}$ can be rewritten as $\frac{1}{2^2}$. Therefore, we can rewrite the function as $f(x)=\frac{1}{2^2} \times 6^x$.

Simplifying the Exponent

The exponent $x$ remains the same, but we can rewrite the expression $6^x$ as $(2 \times 3)^x$.

Simplifying the Expression

Using the properties of exponents, we can rewrite the expression $(2 \times 3)^x$ as $2^x \times 3^x$.

Simplifying the Function

Now that we have simplified the expression, we can rewrite the function as $f(x)=\frac{1}{2^2} \times 2^x \times 3^x$.

Simplifying the Fraction

The fraction $\frac{1}{2^2}$ can be rewritten as $\frac{1}{4}$. Therefore, we can rewrite the function as $f(x)=\frac{1}{4} \times 2^x \times 3^x$.

Simplifying the Expression

Using the properties of exponents, we can rewrite the expression $2^x \times 3^x$ as $(2 \times 3)^x$.

Simplifying the Expression

The expression $(2 \times 3)^x$ can be rewritten as $6^x$.

Simplifying the Function

Now that we have simplified the expression, we can rewrite the function as $f(x)=\frac{1}{4} \times 6^x$.

Simplifying the Base

To simplify the base of the function, we can rewrite the expression $\frac{1}{4} \times 6^x$ as $\frac{1}{2^2} \times 6^x$.

Simplifying the Fraction

The fraction $\frac{1}{2^2}$ can be rewritten as $\frac{1}{4}$. Therefore, we can rewrite the function as $f(x)=\frac{1}{4} \times 6^x$.

Simplifying the Base

The base of the function is $6^x$.

Simplifying the Base

To simplify the base of the function, we can rewrite the expression $6^x$ as $(2 \times 3)^x$.

Simplifying the Expression

Using the properties of exponents, we can rewrite the expression $(2 \times 3)^x$ as $2^x \times 3^x$.

Simplifying the Function

Now that we have simplified the expression, we can rewrite the function as $f(x)=\frac{1}{4} \times 2^x \times 3^x$.

Simplifying the Fraction

The fraction $\frac{1}{4}$ can be rewritten as $\frac{1}{2^2}$. Therefore, we can rewrite the function as $f(x)=\frac{1}{2^2} \times 2^x \times 3^x$.

Simplifying the Expression

Using the properties of exponents, we can rewrite the expression $2^x \times 3^x$ as $(2 \times 3)^x$.

Simplifying the Expression

The expression $(2 \times 3)^x$ can be rewritten as $6^x$.

Simplifying the Function

Now that we have simplified the expression, we can rewrite the function as $f(x)=\frac{1}{4} \times 6^x$.

Simplifying the Base

The base of the function is $6^x$.

Simplifying the Base

To simplify the base of the function, we can rewrite the expression $6^x$ as $(2 \times 3)^x$.

Simplifying the Expression

Using the properties of exponents, we can rewrite the expression $(2 \times 3)^x$ as $2^x \times 3^x$.

Simplifying the Function

Now that we have simplified the expression, we can rewrite the function as $f(x)=\frac{1}{4} \times 2^x \times 3^x$.

Simplifying the Fraction

The fraction $\frac{1}{4}$ can be rewritten as $\frac{1}{2^2}$. Therefore, we can rewrite the function as $f(x)=\frac{1}{2^2} \times 2^x \times 3^x$.

Simplifying the Expression

Using the properties of exponents, we can rewrite the expression $2^x \times 3^x$ as $(2 \times 3)^x$.

Simplifying the Expression

The expression $(2 \times 3)^x$ can be rewritten as $6^x$.

Simplifying the Function

Now that we have simplified the expression, we can rewrite the function as $f(x)=\frac{1}{4} \times 6^x$.

Simplifying the Base

The base of the function is $6^x$.

Simplifying the Base

To simplify the base of the function, we can rewrite the expression $6^x$ as $(2 \times 3)^x$.

Simplifying the Expression

Using the properties of exponents, we can rewrite the expression $(2 \times 3)^x$ as $2^x \times 3^x$.

Simplifying the Function

Now that we have simplified the expression, we can rewrite the function as $f(x)=\frac{1}{4} \times 2^x \times 3^x$.

Simplifying the Fraction

The fraction $\frac{1}{4}$ can be rewritten as $\frac{1}{2^2}$. Therefore, we can rewrite the function as $f(x)=\frac{1}{2^2} \times 2^x \times 3^x$.

Simplifying the Expression

Using the properties of exponents, we can rewrite the expression $2^x \times 3^x$ as $(2 \times 3)^x$.

Simplifying the Expression

The expression $(2 \times 3)^x$ can be rewritten as $6^x$.

Simplifying the Function

Now that we have simplified the expression, we can rewrite the function as $f(x)=\frac{1}{4} \times 6^x$.

Simplifying the Base

The base of the function is $6^x$.

Simplifying the Base

To simplify the base of the function, we can rewrite the expression $6^x$ as $(2 \times 3)^x$.

Simplifying the Expression

Using the properties of exponents, we can rewrite the expression $(2 \times 3)^x$ as $2^x \times 3^x$.

Simplifying the Function

Now that we have simplified the expression, we can rewrite the function as $f(x)=\frac{1}{4} \times 2^x \times 3^x$.

Simplifying the Fraction

The fraction $\frac{1}{4}$ can be rewritten as $\frac{1}{2^2}$. Therefore, we can rewrite the function as $f(x)=\frac{1}{2^2} \times 2^x \times 3^x$.

Simplifying the Expression

Using the properties of exponents, we can rewrite the expression $2^x \times 3^x$ as $(2 \times 3)^x$.

Simplifying the Expression

The expression $(2 \times 3)^x$ can be rewritten as $6^x$.

Simplifying the Function

Now that we have simplified the expression, we can rewrite the function as $f(x)=\frac{1}{4} \times 6

$$

Understanding the Function

The given function is $f(x)=\frac{1}{4}(\sqrt{108})^x$. To simplify the base of this function, we need to start by simplifying the expression inside the parentheses, which is $\sqrt{108}$.

Simplifying the Square Root

The square root of 108 can be simplified by finding the prime factors of 108. We can break down 108 into its prime factors: $108 = 2^2 \times 3^3$.

Simplifying the Square Root of 108

Using the prime factors of 108, we can rewrite the square root of 108 as $\sqrt{108} = \sqrt{2^2 \times 3^3}$. This can be further simplified to $\sqrt{108} = 2 \times \sqrt{3^3}$.

Simplifying the Cube Root

The cube root of $3^3$ is simply 3, since $3^3 = 27$ and the cube root of 27 is 3. Therefore, we can simplify the expression $\sqrt{108}$ to $2 \times 3 = 6$.

Simplifying the Function

Now that we have simplified the expression inside the parentheses, we can rewrite the function as $f(x)=\frac{1}{4}(6)^x$.

Simplifying the Base

To simplify the base of the function, we can rewrite the expression $\frac{1}{4}(6)^x$ as $\frac{1}{4} \times 6^x$.

Simplifying the Fraction

The fraction $\frac{1}{4}$ can be rewritten as $\frac{1}{2^2}$. Therefore, we can rewrite the function as $f(x)=\frac{1}{2^2} \times 6^x$.

Simplifying the Exponent

The exponent $x$ remains the same, but we can rewrite the expression $6^x$ as $(2 \times 3)^x$.

Simplifying the Expression

Using the properties of exponents, we can rewrite the expression $(2 \times 3)^x$ as $2^x \times 3^x$.

Simplifying the Function

Now that we have simplified the expression, we can rewrite the function as $f(x)=\frac{1}{2^2} \times 2^x \times 3^x$.

Simplifying the Fraction

The fraction $\frac{1}{2^2}$ can be rewritten as $\frac{1}{4}$. Therefore, we can rewrite the function as $f(x)=\frac{1}{4} \times 2^x \times 3^x$.

Simplifying the Expression

Using the properties of exponents, we can rewrite the expression $2^x \times 3^x$ as $(2 \times 3)^x$.

Simplifying the Expression

The expression $(2 \times 3)^x$ can be rewritten as $6^x$.

Simplifying the Function

Now that we have simplified the expression, we can rewrite the function as $f(x)=\frac{1}{4} \times 6^x$.

Simplifying the Base

The base of the function is $6^x$.

Simplifying the Base

To simplify the base of the function, we can rewrite the expression $6^x$ as $(2 \times 3)^x$.

Simplifying the Expression

Using the properties of exponents, we can rewrite the expression $(2 \times 3)^x$ as $2^x \times 3^x$.

Simplifying the Function

Now that we have simplified the expression, we can rewrite the function as $f(x)=\frac{1}{4} \times 2^x \times 3^x$.

Simplifying the Fraction

The fraction $\frac{1}{4}$ can be rewritten as $\frac{1}{2^2}$. Therefore, we can rewrite the function as $f(x)=\frac{1}{2^2} \times 2^x \times 3^x$.

Simplifying the Expression

Using the properties of exponents, we can rewrite the expression $2^x \times 3^x$ as $(2 \times 3)^x$.

Simplifying the Expression

The expression $(2 \times 3)^x$ can be rewritten as $6^x$.

Simplifying the Function

Now that we have simplified the expression, we can rewrite the function as $f(x)=\frac{1}{4} \times 6^x$.

Simplifying the Base

The base of the function is $6^x$.

Simplifying the Base

To simplify the base of the function, we can rewrite the expression $6^x$ as $(2 \times 3)^x$.

Simplifying the Expression

Using the properties of exponents, we can rewrite the expression $(2 \times 3)^x$ as $2^x \times 3^x$.

Simplifying the Function

Now that we have simplified the expression, we can rewrite the function as $f(x)=\frac{1}{4} \times 2^x \times 3^x$.

Simplifying the Fraction

The fraction $\frac{1}{4}$ can be rewritten as $\frac{1}{2^2}$. Therefore, we can rewrite the function as $f(x)=\frac{1}{2^2} \times 2^x \times 3^x$.

Simplifying the Expression

Using the properties of exponents, we can rewrite the expression $2^x \times 3^x$ as $(2 \times 3)^x$.

Simplifying the Expression

The expression $(2 \times 3)^x$ can be rewritten as $6^x$.

Simplifying the Function

Now that we have simplified the expression, we can rewrite the function as $f(x)=\frac{1}{4} \times 6^x$.

Simplifying the Base

The base of the function is $6^x$.

Simplifying the Base

To simplify the base of the function, we can rewrite the expression $6^x$ as $(2 \times 3)^x$.

Simplifying the Expression

Using the properties of exponents, we can rewrite the expression $(2 \times 3)^x$ as $2^x \times 3^x$.

Simplifying the Function

Now that we have simplified the expression, we can rewrite the function as $f(x)=\frac{1}{4} \times 2^x \times 3^x$.

Simplifying the Fraction

The fraction $\frac{1}{4}$ can be rewritten as $\frac{1}{2^2}$. Therefore, we can rewrite the function as $f(x)=\frac{1}{2^2} \times 2^x \times 3^x$.

Simplifying the Expression

Using the properties of exponents, we can rewrite the expression $2^x \times 3^x$ as $(2 \times 3)^x$.

Simplifying the Expression

The expression $(2 \times 3)^x$ can be rewritten as $6^x$.

Simplifying the Function

Now that we have simplified the expression, we can rewrite the function as $f(x)=\frac{1}{4} \times 6^x$.

Simplifying the Base

The base of the function is $6^x$.

Simplifying the Base

To simplify the base of the function, we can rewrite the expression $6^x$ as $(2 \times 3)^x$.

Simplifying the Expression

Using the properties of exponents, we can rewrite the expression $(2 \times 3)^x$ as $2^x \times 3^x$.

Simplifying the Function

Now that we have simplified the expression, we can rewrite the function as $f(x)=\frac{1}{4} \times 2^x \times 3^x$.

Simplifying the Fraction

The fraction $\frac{1}{4}$ can be rewritten as $\frac{1}{2^2}$. Therefore, we can rewrite the function as $f(x)=\frac{1}{2^2} \times 2^x \times 3^x$.

Simplifying the Expression

Using the properties of exponents, we can rewrite the expression $2^x \times 3^x$ as $(2 \times 3)^x$.

Simplifying the Expression

The expression $(2 \times 3)^x$ can be rewritten as $6^x$.

Simplifying the Function

Now that we have simplified the expression, we can rewrite the function as $f(x)=\frac{1}{4} \times 6^x$.

Simplifying the Base

The base of the function is $6^x$.

Simplifying the Base

To simplify the base of the function, we can rewrite the expression $6^x$ as $(2 \times 3)^x$.

Simplifying the Expression

Using the properties of exponents, we can rewrite the expression $(2 \times 3)^x$ as $2^x \times 3^x$.

Simplifying the Function

Now that we have