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Difficulty: MediumDecimals and Scientific Notation

If A=0.00048×10−3A = 0.00048 \times 10^{-3} and B=1.2×10−5B = 1.2 \times 10^{-5}, what is the value of A+B4×10−8\frac{A + B}{4 \times 10^{-8}} expressed in scientific notation?

  1. 3.12×1023.12 \times 10^2Answer
  2. B
    1.5×1031.5 \times 10^3
  3. C
    4.2×1024.2 \times 10^2
  4. D
    3.12×10−43.12 \times 10^{-4}
  5. E
    3.12×1013.12 \times 10^1

Answer

3.12×1023.12 \times 10^2
Converting A=0.00048×10−3A = 0.00048 \times 10^{-3} to powers of 10 gives 4.8×10−74.8 \times 10^{-7}, which equals 0.048×10−50.048 \times 10^{-5}. Adding B=1.2×10−5B = 1.2 \times 10^{-5} yields (0.048+1.2)×10−5=1.248×10−5(0.048 + 1.2) \times 10^{-5} = 1.248 \times 10^{-5}. Dividing by 4×10−84 \times 10^{-8} gives 1.2484×10−5−(−8)=0.312×103=3.12×102\frac{1.248}{4} \times 10^{-5 - (-8)} = 0.312 \times 10^3 = 3.12 \times 10^2.

Step-by-Step Solution

1
Express AA in standard scientific notation and then match its exponent to BB's exponent.
A=0.00048×10−3=4.8×10−7=0.048×10−5A = 0.00048 \times 10^{-3} = 4.8 \times 10^{-7} = 0.048 \times 10^{-5}
To perform addition between numbers in scientific notation, their exponents must be equal.
2
Add AA and BB.
A+B=0.048×10−5+1.2×10−5=1.248×10−5A + B = 0.048 \times 10^{-5} + 1.2 \times 10^{-5} = 1.248 \times 10^{-5}
Combine the coefficients once the powers of 10 match.
3
Divide A+BA + B by 4×10−84 \times 10^{-8} and express the final result in scientific notation.
1.248×10−54×10−8=(1.2484)×10−5−(−8)=0.312×103=3.12×102\frac{1.248 \times 10^{-5}}{4 \times 10^{-8}} = \left(\frac{1.248}{4}\right) \times 10^{-5 - (-8)} = 0.312 \times 10^3 = 3.12 \times 10^2
Divide the coefficients and subtract the exponent in the denominator from the exponent in the numerator.

Key Concept

Decimals, Place Value, and Operations in Scientific Notation
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Topics

Decimals and Scientific Notation

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