Without Using a calculator, work out (9.15 x 10-¹2) (4.23 × 10-5)
Give your answer in standard form to 3 significant figures.
Answer
The Correct Answer and Explanation is:
To solve the expression (9.15×10−12)(4.23×10−5)(9.15 \times 10^{-12})(4.23 \times 10^{-5})(9.15×10−12)(4.23×10−5) and express the result in standard form to three significant figures, we can break it down into a few steps:
Step 1: Multiply the significant figures.
The first part of the expression involves multiplying the decimal numbers:9.15×4.239.15 \times 4.239.15×4.23
To simplify, we can estimate this multiplication:
- 9.15≈99.15 \approx 99.15≈9 and 4.23≈44.23 \approx 44.23≈4
- Multiply 9×4=369 \times 4 = 369×4=36
However, to be more precise, let’s calculate it without rounding:9.15×4.23=38.83459.15 \times 4.23 = 38.83459.15×4.23=38.8345
Step 2: Add the exponents.
The second part of the expression involves multiplying the powers of 10:10−12×10−510^{-12} \times 10^{-5}10−12×10−5
When multiplying powers of 10, you add the exponents:10−12×10−5=10−12+(−5)=10−1710^{-12} \times 10^{-5} = 10^{-12 + (-5)} = 10^{-17}10−12×10−5=10−12+(−5)=10−17
Step 3: Combine the results.
Now, we combine the result from Step 1 with the result from Step 2:(9.15×4.23)×10−12×10−5=38.8345×10−17(9.15 \times 4.23) \times 10^{-12} \times 10^{-5} = 38.8345 \times 10^{-17}(9.15×4.23)×10−12×10−5=38.8345×10−17
Step 4: Convert to standard form.
To express the result in standard form, we need the decimal number to be between 1 and 10.38.8345=3.88345×10138.8345 = 3.88345 \times 10^138.8345=3.88345×101
So:38.8345×10−17=3.88345×101×10−17=3.88345×10−1638.8345 \times 10^{-17} = 3.88345 \times 10^1 \times 10^{-17} = 3.88345 \times 10^{-16}38.8345×10−17=3.88345×101×10−17=3.88345×10−16
Step 5: Round to three significant figures.
Finally, we round the result to three significant figures:3.88345≈3.883.88345 \approx 3.883.88345≈3.88
Thus, the answer in standard form to three significant figures is:3.88×10−16\boxed{3.88 \times 10^{-16}}3.88×10−16
Summary:
- Multiply the significant figures: 9.15×4.23=38.83459.15 \times 4.23 = 38.83459.15×4.23=38.8345.
- Add the exponents: 10−12×10−5=10−1710^{-12} \times 10^{-5} = 10^{-17}10−12×10−5=10−17.
- Combine the two parts: 38.8345×10−17=3.88×10−1638.8345 \times 10^{-17} = 3.88 \times 10^{-16}38.8345×10−17=3.88×10−16.
- Round the result to three significant figures: 3.88×10−163.88 \times 10^{-16}3.88×10−16.
The final result is 3.88×10−163.88 \times 10^{-16}3.88×10−16.