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Difficulty: MediumDecimals and Scientific Notation

A high-precision optical sensor measures a time interval as T=0.00064×(2.5×108)1.6×10−1T = \frac{0.00064 \times (2.5 \times 10^8)}{1.6 \times 10^{-1}} nanoseconds. Which of the following values are equivalent to TT? Select all such values.

  1. 1.0×1061.0 \times 10^6Answer
  2. B
    100,000100,000
  3. 4.0×10740\frac{4.0 \times 10^7}{40}Answer
  4. (2.5×103)×(4.0×102)(2.5 \times 10^3) \times (4.0 \times 10^2)Answer
  5. E
    1.0×10−61.0 \times 10^{-6}

Answer

The values equivalent to TT are 1.0×1061.0 \times 10^6, 4.0×10740\frac{4.0 \times 10^7}{40}, and (2.5×103)×(4.0×102)(2.5 \times 10^3) \times (4.0 \times 10^2).
Evaluating TT gives (6.4×10−4)×(2.5×108)1.6×10−1=1.6×1051.6×10−1=1.0×106\frac{(6.4 \times 10^{-4}) \times (2.5 \times 10^8)}{1.6 \times 10^{-1}} = \frac{1.6 \times 10^5}{1.6 \times 10^{-1}} = 1.0 \times 10^6. The expression representing 1.0×1061.0 \times 10^6 directly matches TT. The quotient 4.0×10740\frac{4.0 \times 10^7}{40} simplifies to 4.0×1074.0×101=1.0×106\frac{4.0 \times 10^7}{4.0 \times 10^1} = 1.0 \times 10^6, which matches TT. The product (2.5×103)×(4.0×102)(2.5 \times 10^3) \times (4.0 \times 10^2) simplifies to 10.0×105=1.0×10610.0 \times 10^5 = 1.0 \times 10^6, which also matches TT.

Step-by-Step Solution

1
Convert the decimal 0.000640.00064 into scientific notation
0.00064=6.4×10−40.00064 = 6.4 \times 10^{-4}
Expressing all terms in scientific notation simplifies exponent operations.
2
Simplify the numerator of the expression for TT
(6.4×10−4)×(2.5×108)=(6.4×2.5)×10−4+8=16.0×104=1.6×105(6.4 \times 10^{-4}) \times (2.5 \times 10^8) = (6.4 \times 2.5) \times 10^{-4 + 8} = 16.0 \times 10^4 = 1.6 \times 10^5
Multiply coefficients directly and add exponents for product of powers with equal base.
3
Divide the numerator by the denominator 1.6×10−11.6 \times 10^{-1}
T=1.6×1051.6×10−1=(1.61.6)×105−(−1)=1.0×106=1,000,000T = \frac{1.6 \times 10^5}{1.6 \times 10^{-1}} = \left(\frac{1.6}{1.6}\right) \times 10^{5 - (-1)} = 1.0 \times 10^6 = 1,000,000
Subtract the denominator exponent from the numerator exponent: 5−(−1)=65 - (-1) = 6.
4
Evaluate the choices to check equivalence to 1.0×1061.0 \times 10^6
1.0×1061.0 \times 10^6, 4.0×10740=1.0×106\frac{4.0 \times 10^7}{40} = 1.0 \times 10^6, and (2.5×103)×(4.0×102)=10.0×105=1.0×106(2.5 \times 10^3) \times (4.0 \times 10^2) = 10.0 \times 10^5 = 1.0 \times 10^6 are all equal to TT.
Matching each simplified expression to the calculated value of TT determines the correct options.

Key Concept

Operations with Decimals and Exponents in Scientific Notation
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Topics

Decimals and Scientific Notation

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