(3.5 × 10??) (3 × 10?¹?) Express your answer in scientific notation. 1.05 × 10?¹? 1.05 × 10?¹? 1.05 × 10?¹³ 1.05 × 10¹?
The Correct Answer and Explanation is:
Correct Answer:
1.05 × 10⁻¹³
Explanation
Scientific notation is a method of expressing numbers that are too large or too small to be conveniently written in decimal form. It uses powers of 10 to simplify these expressions. In this case, the problem involves multiplying two numbers written in scientific notation: (3.5×10?)×(3×10−14)(3.5 × 10^?) × (3 × 10^{-14})(3.5×10?)×(3×10−14)
Assume the missing exponent in the first number is 1, as suggested by the answer options. This makes the expression: (3.5×101)×(3×10−14)(3.5 × 10^1) × (3 × 10^{-14})(3.5×101)×(3×10−14)
Multiplication in scientific notation follows two steps:
- Multiply the decimal numbers: 3.5×3=10.53.5 × 3 = 10.53.5×3=10.5
- Add the exponents of 10: 101×10−14=10(1+(−14))=10−1310^1 × 10^{-14} = 10^{(1 + (-14))} = 10^{-13}101×10−14=10(1+(−14))=10−13
So the result becomes: 10.5×10−1310.5 × 10^{-13}10.5×10−13
However, in scientific notation, the number before the power of 10 should be between 1 and 10. Since 10.5 is greater than 10, it needs to be adjusted. Dividing 10.5 by 10 gives 1.05, and increasing the exponent by 1 keeps the value unchanged: 10.5×10−13=1.05×10−1210.5 × 10^{-13} = 1.05 × 10^{-12}10.5×10−13=1.05×10−12
But none of the choices show 1.05×10−121.05 × 10^{-12}1.05×10−12. Let’s recheck the assumed exponent in the first number. Try assuming the first exponent is -1 instead: (3.5×10−1)×(3×10−14)=10.5×10−15=1.05×10−14(3.5 × 10^{-1}) × (3 × 10^{-14}) = 10.5 × 10^{-15} = 1.05 × 10^{-14}(3.5×10−1)×(3×10−14)=10.5×10−15=1.05×10−14
Still doesn’t match.
Now try exponent 0: (3.5×100)×(3×10−14)=10.5×10−14=1.05×10−13(3.5 × 10^0) × (3 × 10^{-14}) = 10.5 × 10^{-14} = 1.05 × 10^{-13}(3.5×100)×(3×10−14)=10.5×10−14=1.05×10−13
Now the expression fits a valid scientific format and matches an answer choice.
Thus, the correct answer is:
1.05 × 10⁻¹³.
