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Simplify the expression. Write the answer using scientific notation.

[tex]\[
\left(2 \times 10^{13}\right)^{-2}
\][/tex]

A. [tex]\(0.25 \times 10^{-26}\)[/tex]

B. [tex]\(-4 \times 10^{-26}\)[/tex]

C. [tex]\(2.5 \times 10^{-27}\)[/tex]

D. [tex]\(2.5 \times 10^{-26}\)[/tex]


Sagot :

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].