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Difficulty: HardDecimals and Scientific Notation

Let A=0.000032×10nA = 0.000032 \times 10^{n} and B=8.0×10n−3B = 8.0 \times 10^{n-3}, where nn is an integer. If A2B=1.28×10−4\frac{A^2}{B} = 1.28 \times 10^{-4}, what is the value of nn?

  1. A
    −2-2
  2. B
    11
  3. 33Answer
  4. D
    44
  5. E
    99

Answer

The value of nn is 33.
Rewriting 0.0000320.000032 as 3.2×10−53.2 \times 10^{-5} allows AA to be expressed as 3.2×10n−53.2 \times 10^{n-5}. Squaring AA yields 10.24×102n−1010.24 \times 10^{2n-10}. Dividing by B=8.0×10n−3B = 8.0 \times 10^{n-3} gives 1.28×10n−71.28 \times 10^{n-7}. Equating n−7=−4n - 7 = -4 directly leads to n=3n = 3.

Step-by-Step Solution

1
Convert AA to standard scientific notation in terms of nn
A=3.2×10−5×10n=3.2×10n−5A = 3.2 \times 10^{-5} \times 10^n = 3.2 \times 10^{n-5}
The decimal 0.0000320.000032 equals 3.2×10−53.2 \times 10^{-5} because the decimal point is shifted 55 places to the right.
2
Calculate A2A^2
A2=(3.2×10n−5)2=(3.2)2×102(n−5)=10.24×102n−10A^2 = (3.2 \times 10^{n-5})^2 = (3.2)^2 \times 10^{2(n-5)} = 10.24 \times 10^{2n-10}
Apply the power of a product rule (ab)k=akbk(ab)^k = a^k b^k and exponent power rule (10p)q=10pq(10^p)^q = 10^{pq}.
3
Divide A2A^2 by BB
\frac{A^2}{B} = \frac{10.24 \times 10^{2n-10}}{8.0 \times 10^{n-3}} = \left(\frac{10.24}{8.0}\right) \times 10^{(2n-10) - (n-3)} = 1.28 \times 10^{n-7}
Divide coefficients (10.24/8.0=1.28)(10.24 / 8.0 = 1.28) and subtract powers of ten exponents ((2n−10)−(n−3)=n−7)((2n - 10) - (n - 3) = n - 7).
4
Equate the simplified expression to the given value and solve for nn
1.28×10n−7=1.28×10−4  ⟹  n−7=−4  ⟹  n=31.28 \times 10^{n-7} = 1.28 \times 10^{-4} \implies n - 7 = -4 \implies n = 3
Since the coefficients match (1.281.28), equate the exponents of 1010 to solve for nn.

Key Concept

Scientific Notation Operations and Exponent Laws
Estimated Time:2m 0s
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Topics

Decimals and Scientific Notation

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