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Difficulty: HardDecimals and Scientific Notation

Consider the expression N=0.00048×(2.5×107)1.2×10−2N = \frac{0.00048 \times (2.5 \times 10^7)}{1.2 \times 10^{-2}}. Which of the following expressions are equal to NN? Select all such values.

  1. 1.0×1061.0 \times 10^6Answer
  2. (4.0×104)×25(4.0 \times 10^4) \times 25Answer
  3. C
    1.0×1021.0 \times 10^2
  4. 5.0×10−35.0×10−9\frac{5.0 \times 10^{-3}}{5.0 \times 10^{-9}}Answer
  5. E
    0.00012×1090.00012 \times 10^9

Answer

The expressions equal to NN are 1.0×1061.0 \times 10^6, (4.0×104)×25(4.0 \times 10^4) \times 25, and 5.0×10−35.0×10−9\frac{5.0 \times 10^{-3}}{5.0 \times 10^{-9}}.
Evaluating NN simplifies to 1.0×1061.0 \times 10^6. The expression stating 1.0×1061.0 \times 10^6 is an exact scientific notation match. The expression (4.0×104)×25(4.0 \times 10^4) \times 25 computes to 40,000×25=1,000,00040,000 \times 25 = 1,000,000. The fraction 5.0×10−35.0×10−9\frac{5.0 \times 10^{-3}}{5.0 \times 10^{-9}} evaluates to 1.0×10−3−(−9)=1.0×1061.0 \times 10^{-3 - (-9)} = 1.0 \times 10^6.

Step-by-Step Solution

1
Express all numbers in the stem's numerator in standard scientific notation
0.00048=4.8×10−40.00048 = 4.8 \times 10^{-4} and the numerator becomes (4.8×10−4)×(2.5×107)(4.8 \times 10^{-4}) \times (2.5 \times 10^7)
Standardizing numbers into scientific notation simplifies operations involving powers of 10
2
Multiply the coefficients and combine the powers of 10 in the numerator
(4.8×2.5)×10−4+7=12×103=1.2×104(4.8 \times 2.5) \times 10^{-4 + 7} = 12 \times 10^3 = 1.2 \times 10^4
Coefficients are multiplied directly while exponents are added according to index laws
3
Divide the simplified numerator by the denominator
1.2×1041.2×10−2=(1.21.2)×104−(−2)=1.0×106=1,000,000\frac{1.2 \times 10^4}{1.2 \times 10^{-2}} = \left(\frac{1.2}{1.2}\right) \times 10^{4 - (-2)} = 1.0 \times 10^6 = 1,000,000
Subtracting a negative exponent in the denominator is equivalent to adding its positive value
4
Evaluate each given option to determine equivalence to 1.0×1061.0 \times 10^6
The expressions representing 1.0×1061.0 \times 10^6 are 1.0×1061.0 \times 10^6, (4.0×104)×25(4.0 \times 10^4) \times 25, and 5.0×10−35.0×10−9\frac{5.0 \times 10^{-3}}{5.0 \times 10^{-9}}
Direct comparison verifies which options match the calculated value of NN

Key Concept

Operations with Decimals and Scientific Notation
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Topics

Decimals and Scientific Notation

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