To solve the expression [tex]\(\left(2 \times 10^{13}\right)^{-2}\)[/tex], let's work through the steps systematically.
1. Understand the Expression:
The given expression is [tex]\(\left(2 \times 10^{13}\right)^{-2}\)[/tex].
2. Simplify the Expression Inside the Parentheses:
The term inside the parentheses is [tex]\(2 \times 10^{13}\)[/tex]. We need to square this term and invert it since it is raised to the power of [tex]\(-2\)[/tex].
3. Square the Term:
When squaring the term inside the parentheses, [tex]\((2 \times 10^{13})^2\)[/tex], we need to square both parts:
- The constant 2 squared is [tex]\(2^2 = 4\)[/tex].
- For the exponent part, [tex]\((10^{13})^2 = 10^{26}\)[/tex].
So, [tex]\((2 \times 10^{13})^2 = 4 \times 10^{26}\)[/tex].
4. Take the Inverse:
Since we have the term squared, now we need to take the reciprocal because it was initially raised to the power of [tex]\(-2\)[/tex]:
[tex]\[
\left(4 \times 10^{26}\right)^{-1} = \frac{1}{4 \times 10^{26}}
\][/tex]
5. Simplify the Reciprocal:
To simplify this, we can rewrite:
[tex]\[
\frac{1}{4 \times 10^{26}} = \frac{1}{4} \times 10^{-26}
\][/tex]
Here, [tex]\(\frac{1}{4} = 0.25\)[/tex].
6. Express in Scientic Notation:
[tex]\[
0.25 \times 10^{-26}
\][/tex]
Now let's convert [tex]\(0.25\)[/tex] into scientific notation. We know:
[tex]\[
0.25 = 2.5 \times 10^{-1}
\][/tex]
When we multiply this by [tex]\(10^{-26}\)[/tex]:
[tex]\[
2.5 \times 10^{-1} \times 10^{-26} = 2.5 \times 10^{-27}
\][/tex]
7. Verify:
At this point, we need to compare our result with the given options:
- [tex]\(0.25 \times 10^{-26}\)[/tex]
- [tex]\(-4 \times 10^{-26}\)[/tex]
- [tex]\(2.5 \times 10^{-27}\)[/tex]
- [tex]\(2.5 \times 10^{-26}\)[/tex]
The simplified expression is correctly matched to [tex]\(2.5 \times 10^{-27}\)[/tex].
So, the final answer is [tex]\(2.5 \times 10^{-27}\)[/tex]. Thus, the correct option is:
[tex]\(2.5 \times 10^{-27}\)[/tex].
