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Difficulty: MediumDecimals and Scientific Notation

Two deep-space radio signals have power measurements of S1=4.5×10−7S_1 = 4.5 \times 10^{-7} watts and S2=3.5×10−6S_2 = 3.5 \times 10^{-6} watts. A combined signal power is defined as S3=S1+S2S_3 = S_1 + S_2. When S3S_3 is written in scientific notation as a×10na \times 10^n, where 1≤a<101 \leq a < 10 and nn is an integer, what is the value of a+na + n?

  1. A
    −5.00-5.00
  2. −2.05-2.05Answer
  3. C
    2.002.00
  4. D
    9.959.95
  5. E
    32.5032.50

Answer

The value of a+na + n is −2.05-2.05.
To add numbers in scientific notation, first rewrite them with identical powers of 10. Expressing 4.5×10−74.5 \times 10^{-7} as 0.45×10−60.45 \times 10^{-6} allows direct addition with 3.5×10−63.5 \times 10^{-6}, resulting in 3.95×10−63.95 \times 10^{-6}. Here a=3.95a = 3.95 (which satisfies 1≤a<101 \leq a < 10) and n=−6n = -6. Adding a+na + n yields 3.95+(−6)=−2.053.95 + (-6) = -2.05.

Step-by-Step Solution

1
Express both quantities with a common power of 10 to allow addition.
S1=4.5×10−7=0.45×10−6S_1 = 4.5 \times 10^{-7} = 0.45 \times 10^{-6} watts.
Before adding numbers in scientific notation, their powers of 10 must match.
2
Add the coefficients while maintaining the common power of 10.
S3=(0.45+3.5)×10−6=3.95×10−6S_3 = (0.45 + 3.5) \times 10^{-6} = 3.95 \times 10^{-6} watts.
Distributive property allows adding coefficients once exponents match.
3
Verify proper scientific notation form a×10na \times 10^n where 1≤a<101 \leq a < 10.
a=3.95a = 3.95 and n=−6n = -6.
The coefficient 3.953.95 satisfies 1≤3.95<101 \leq 3.95 < 10, so no further decimal shift is needed.
4
Calculate the requested sum a+na + n.
3.95+(−6)=−2.053.95 + (-6) = -2.05.
Adding the coefficient 3.953.95 to the negative exponent −6-6 gives −2.05-2.05.

Key Concept

Scientific Notation Addition and Place Value Alignment
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Topics

Decimals and Scientific Notation

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