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Difficulty: HardScientific Notation

When the expression (8.0×104)×(6.0×10−7)1.5×103\frac{(8.0 \times 10^4) \times (6.0 \times 10^{-7})}{1.5 \times 10^3} is evaluated and the result is written in standard scientific notation as a×10na \times 10^n, where 1≤a<101 \le a < 10, what is the value of the integer nn?

Answer: -5

Answer

The exponent is -5.
Evaluating the numerator yields 48×10−348 \times 10^{-3}. Dividing this by the denominator 1.5×1031.5 \times 10^3 results in 32×10−632 \times 10^{-6}. To write this in standard scientific notation, the coefficient must be strictly greater than or equal to 1 and strictly less than 10, so 32 is rewritten as 3.2×1013.2 \times 10^1. Applying this shift makes the final expression 3.2×10−53.2 \times 10^{-5}. Thus, the final exponent nn is -5.

Step-by-Step Solution

1
Multiply the expressions in the numerator
48×10−348 \times 10^{-3}
Following the order of operations, simplify the numerator first by multiplying the coefficients (8.0×6.0=488.0 \times 6.0 = 48) and adding the exponents of the base 10 (4+(−7)=−34 + (-7) = -3).
2
Divide by the denominator
32×10−632 \times 10^{-6}
Divide the resulting numerator's coefficient by the denominator's coefficient (48÷1.5=3248 \div 1.5 = 32) and subtract the denominator's exponent from the numerator's exponent (−3−3=−6-3 - 3 = -6).
3
Normalize the coefficient to standard scientific notation
3.2×10−53.2 \times 10^{-5}
Standard scientific notation requires the coefficient (aa) to be between 1 and 10. We rewrite 3232 as 3.2×1013.2 \times 10^1. Multiplying this by the existing 10−610^{-6} changes the overall exponent to 1+(−6)=−51 + (-6) = -5.
4
Identify the final value of nn
-5
The question asks for the integer nn from the format a×10na \times 10^n. Comparing our final result 3.2×10−53.2 \times 10^{-5} to this format, n=−5n = -5.

Key Concept

Operations with scientific notation and coefficient normalization
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Topics

Scientific Notation

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