Sure! Let’s find the solutions for the given exponential expressions step by step.
### Part b
Consider the expression:
[tex]\[
\left(\frac{1}{27}\right)^2 \times 3^2 \times \frac{1}{3}
\][/tex]
1. Evaluate [tex]\(\left(\frac{1}{27}\right)^2\)[/tex]:
[tex]\[
\left(\frac{1}{27}\right)^2 = \frac{1}{27 \times 27} = \frac{1}{729}
\][/tex]
2. Evaluate [tex]\(3^2\)[/tex]:
[tex]\[
3^2 = 9
\][/tex]
3. Combine the results from steps 1 and 2:
[tex]\[
\frac{1}{729} \times 9 = \frac{9}{729}
\][/tex]
4. Simplify the fraction:
[tex]\[
\frac{9}{729} = \frac{1}{81}
\][/tex]
5. Finally, multiply by [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
\frac{1}{81} \times \frac{1}{3} = \frac{1}{243}
\][/tex]
Therefore, the result for Part b is:
[tex]\[
\left(\frac{1}{27}\right)^2 \times 3^2 \times \frac{1}{3} = 0.004115226337448559
\][/tex]
### Part c
Consider the expression:
[tex]\[
8^2 \times 4^2 : \frac{1}{2}
\][/tex]
1. Evaluate [tex]\(8^2\)[/tex]:
[tex]\[
8^2 = 64
\][/tex]
2. Evaluate [tex]\(4^2\)[/tex]:
[tex]\[
4^2 = 16
\][/tex]
3. Combine the results from steps 1 and 2:
[tex]\[
64 \times 16
\][/tex]
[tex]\[
64 \times 16 = 1024
\][/tex]
4. Divide by [tex]\(\frac{1}{2}\)[/tex] (which is equivalent to multiplying by 2):
[tex]\[
1024 \div \frac{1}{2} = 1024 \times 2
\][/tex]
[tex]\[
1024 \times 2 = 2048
\][/tex]
Therefore, the result for Part c is:
[tex]\[
8^2 \times 4^2 : \frac{1}{2} = 2048.0
\][/tex]
In summary:
- Part b result = [tex]\(0.004115226337448559\)[/tex]
- Part c result = [tex]\(2048.0\)[/tex]
