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Difficulty: MediumScientific Notation

A rectangular microchip has a length of 8.0×10−38.0 \times 10^{-3} meters and a width of 4.5×10−34.5 \times 10^{-3} meters. Calculate the area of the microchip in square meters and express the result in proper scientific notation as a×10na \times 10^n (where 1≤a<101 \le a < 10). What is the value of the coefficient aa?

Answer: 3.6

Answer

The coefficient aa is 3.6.
The correct answer is derived by multiplying the coefficients (8.0 × 4.5 = 36), adding the exponents (-3 + -3 = -6), and normalizing the preliminary result of 36 × 10^-6 by moving the decimal left by one place. This creates a proper coefficient of 3.6 and changes the exponent to -5.

Step-by-Step Solution

1
Set up the multiplication for area
Area = (8.0×10−3)×(4.5×10−3)(8.0 \times 10^{-3}) \times (4.5 \times 10^{-3})
The area of a rectangle is found by multiplying its length by its width.
2
Multiply the coefficients
8.0×4.5=368.0 \times 4.5 = 36
When multiplying terms in scientific notation, the numerical coefficients are multiplied together.
3
Multiply the powers of 10
10−3×10−3=10−610^{-3} \times 10^{-3} = 10^{-6}
When multiplying identical bases, their exponents are added: −3+(−3)=−6-3 + (-3) = -6.
4
Combine into preliminary result
36×10−636 \times 10^{-6}
Bring the multiplied coefficient and combined power of 10 together.
5
Normalize to proper scientific notation
3.6×10−53.6 \times 10^{-5}
Proper scientific notation requires the coefficient aa to satisfy 1≤a<101 \le a < 10. Moving the decimal one place left on 36 increases the exponent by 1 (−6+1=−5-6 + 1 = -5).
6
Extract the coefficient aa
3.6
The question specifically asks for the value of the coefficient aa in the final a×10na \times 10^n format.

Key Concept

Multiplying and Normalizing Values in Scientific Notation
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Topics

Scientific Notation

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