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Difficulty: Very hardDecimals and Scientific Notation

Let M=3.2×10−5M = 3.2 \times 10^{-5} and N=8.0×10−4N = 8.0 \times 10^{-4}. Which of the following statements must be true? Select all that apply.

  1. N−M=7.68×10−4N - M = 7.68 \times 10^{-4}Answer
  2. The ratio NM\frac{N}{M} is equal to 2525Answer
  3. C
    M⋅N=1.6×10−5\sqrt{M \cdot N} = 1.6 \times 10^{-5}
  4. D
    M+N=1.12×10−3M + N = 1.12 \times 10^{-3}
  5. 1M=3.125×104\frac{1}{M} = 3.125 \times 10^4Answer

Answer

The correct statements are N−M=7.68×10−4N - M = 7.68 \times 10^{-4}, the ratio NM\frac{N}{M} is equal to 2525, and 1M=3.125×104\frac{1}{M} = 3.125 \times 10^4.
The statement N−M=7.68×10−4N - M = 7.68 \times 10^{-4} is true because expressing both numbers with the exponent 10−410^{-4} gives (8.0−0.32)×10−4=7.68×10−4(8.0 - 0.32) \times 10^{-4} = 7.68 \times 10^{-4}. The statement that the ratio NM\frac{N}{M} is equal to 2525 is true because 8.03.2×10−4−(−5)=2.5×10=25\frac{8.0}{3.2} \times 10^{-4 - (-5)} = 2.5 \times 10 = 25. The statement 1M=3.125×104\frac{1}{M} = 3.125 \times 10^4 is true because 13.2×105=0.3125×105=3.125×104\frac{1}{3.2} \times 10^5 = 0.3125 \times 10^5 = 3.125 \times 10^4.

Step-by-Step Solution

1
Evaluate N−MN - M
N−M=7.68×10−4N - M = 7.68 \times 10^{-4}
Convert MM to the exponent 10−410^{-4}: M=0.32×10−4M = 0.32 \times 10^{-4}. Then N−M=(8.0−0.32)×10−4=7.68×10−4N - M = (8.0 - 0.32) \times 10^{-4} = 7.68 \times 10^{-4}.
2
Evaluate NM\frac{N}{M}
NM=25\frac{N}{M} = 25
Divide coefficients and subtract exponents: 8.03.2×10−4−(−5)=2.5×101=25\frac{8.0}{3.2} \times 10^{-4 - (-5)} = 2.5 \times 10^1 = 25.
3
Evaluate M⋅N\sqrt{M \cdot N}
\sqrt{M \cdot N} = 1.6 \times 10^{-4}
Compute M⋅N=25.6×10−9=2.56×10−8M \cdot N = 25.6 \times 10^{-9} = 2.56 \times 10^{-8}. Take the square root: 2.56×10−8=1.6×10−4\sqrt{2.56} \times \sqrt{10^{-8}} = 1.6 \times 10^{-4}.
4
Evaluate M+NM + N
M+N=8.32×10−4M + N = 8.32 \times 10^{-4}
Express with common powers of ten: 0.32×10−4+8.0×10−4=8.32×10−40.32 \times 10^{-4} + 8.0 \times 10^{-4} = 8.32 \times 10^{-4}.
5
Evaluate 1M\frac{1}{M}
1M=3.125×104\frac{1}{M} = 3.125 \times 10^4
Compute 13.2×10−5=0.3125×105=3.125×104\frac{1}{3.2 \times 10^{-5}} = 0.3125 \times 10^5 = 3.125 \times 10^4.

Key Concept

Operations with numbers in scientific notation and place value adjustments
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Topics

Decimals and Scientific Notation

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