Question
Calculate \[ \left(2.0 \times 10^{9}\right)^{-2} \] Write your answer in scientific notation.
Ask by Weston Henry. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\(2.5 \times 10^{-19}\)
Solution
Calculate or simplify the expression \( (2.0 \times 10^{9})^{-2} \).
Calculate the value by following steps:
- step0: Calculate:
\(\left(2\times 10^{9}\right)^{-2}\)
- step1: Express with a positive exponent:
\(\frac{1}{\left(2\times 10^{9}\right)^{2}}\)
- step2: Evaluate the power:
\(\frac{1}{4\times 10^{18}}\)
- step3: Separate the fraction:
\(\frac{1}{4}\times \frac{1}{10^{18}}\)
- step4: Divide:
\(0.25\times \frac{1}{10^{18}}\)
- step5: Divide:
\(0.25\times 10^{-18}\)
- step6: Rewrite the number:
\(2.5\times 10^{-19}\)
The result of \((2.0 \times 10^{9})^{-2}\) is \(2.5 \times 10^{-19}\) in scientific notation.
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Simplify this solution Bonus Knowledge
To calculate \(\left(2.0 \times 10^{9}\right)^{-2}\), we first handle the coefficient and the exponent separately. 1. For the coefficient \(2.0^{-2}\): \[ 2.0^{-2} = \frac{1}{(2.0)^2} = \frac{1}{4.0} = 0.25 \] 2. For the exponent \(10^{9}\): \[ (10^{9})^{-2} = 10^{-18} \] Now, we combine these results: \[ \left(2.0 \times 10^{9}\right)^{-2} = 0.25 \times 10^{-18} \] To express \(0.25\) in scientific notation, we rewrite it as \(2.5 \times 10^{-1}\). Therefore, we have: \[ 2.5 \times 10^{-1} \times 10^{-18} = 2.5 \times 10^{-1 - 18} = 2.5 \times 10^{-19} \] The final answer in scientific notation is: \[ \boxed{2.5 \times 10^{-19}} \]
