Let's break this problem down step by step and solve it.
We need to simplify the expression:
[tex]\[
\frac{\left(\frac{1}{3}\right)^4 \times\left(\frac{1}{3}\right)^6}{\left[\left(\frac{1}{3}\right)^4\right]^2}
\][/tex]
### Step 1: Simplify each part individually
#### Numerator
First, look at the numerator:
[tex]\[
\left(\frac{1}{3}\right)^4 \times \left(\frac{1}{3}\right)^6
\][/tex]
We can use the property of exponents that [tex]\(a^m \times a^n = a^{m+n}\)[/tex]:
[tex]\[
\left(\frac{1}{3}\right)^4 \times \left(\frac{1}{3}\right)^6 = \left(\frac{1}{3}\right)^{4+6} = \left(\frac{1}{3}\right)^{10}
\][/tex]
#### Denominator
Next, look at the denominator:
[tex]\[
\left[\left(\frac{1}{3}\right)^4\right]^2
\][/tex]
Using the property of exponents that [tex]\((a^m)^n = a^{m \times n}\)[/tex]:
[tex]\[
\left[\left(\frac{1}{3}\right)^4\right]^2 = \left(\frac{1}{3}\right)^{4 \times 2} = \left(\frac{1}{3}\right)^8
\][/tex]
### Step 2: Combine the simplified parts
Now we need to combine the numerator and the denominator:
[tex]\[
\frac{\left(\frac{1}{3}\right)^{10}}{\left(\frac{1}{3}\right)^8}
\][/tex]
We can use the property of exponents that [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
\frac{\left(\frac{1}{3}\right)^{10}}{\left(\frac{1}{3}\right)^8} = \left(\frac{1}{3}\right)^{10-8} = \left(\frac{1}{3}\right)^2
\][/tex]
### Step 3: Calculate the final result
Calculate [tex]\(\left(\frac{1}{3}\right)^2\)[/tex]:
[tex]\[
\left(\frac{1}{3}\right)^2 = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}
\][/tex]
In decimal form, [tex]\(\frac{1}{9}\)[/tex] is approximately:
[tex]\[
\frac{1}{9} = 0.1111111111111111
\][/tex]
Therefore, the simplified result of the original expression is:
[tex]\[
\boxed{0.1111111111111111}
\][/tex]
