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Difficulty: HardDecimals and Scientific Notation

Let p=0.00036×10−2p = 0.00036 \times 10^{-2} and q=9.0×10−7q = 9.0 \times 10^{-7}. Which of the following statements are true? Select all that apply.

  1. p+q=4.5×10−6p + q = 4.5 \times 10^{-6}Answer
  2. pq=4\frac{p}{q} = 4Answer
  3. p×q=1.8×10−6\sqrt{p \times q} = 1.8 \times 10^{-6}Answer
  4. D
    p−q=2.7×10−7p - q = 2.7 \times 10^{-7}
  5. E
    p2+q2=p+q\sqrt{p^2 + q^2} = p + q

Answer

The correct statements are the ones asserting p+q=4.5×10−6p + q = 4.5 \times 10^{-6}, pq=4\frac{p}{q} = 4, and p×q=1.8×10−6\sqrt{p \times q} = 1.8 \times 10^{-6}.
Converting pp to 3.6×10−63.6 \times 10^{-6} and qq to 0.9×10−60.9 \times 10^{-6} shows that their sum is 4.5×10−64.5 \times 10^{-6}, their ratio is 44, and the square root of their product 3.24×10−12\sqrt{3.24 \times 10^{-12}} is 1.8×10−61.8 \times 10^{-6}.

Step-by-Step Solution

1
Convert pp into standard scientific notation.
p=0.00036×10−2=3.6×10−4×10−2=3.6×10−6p = 0.00036 \times 10^{-2} = 3.6 \times 10^{-4} \times 10^{-2} = 3.6 \times 10^{-6}.
Expressing numbers in consistent scientific notation enables direct arithmetic operations.
2
Align qq to the same power of ten for comparison.
q=9.0×10−7=0.9×10−6q = 9.0 \times 10^{-7} = 0.9 \times 10^{-6}.
Writing terms with matching exponents allows straightforward addition and subtraction.
3
Test each statement using the simplified values.
p+q=(3.6+0.9)×10−6=4.5×10−6p + q = (3.6 + 0.9) \times 10^{-6} = 4.5 \times 10^{-6} (True); pq=3.6×10−60.9×10−6=4\frac{p}{q} = \frac{3.6 \times 10^{-6}}{0.9 \times 10^{-6}} = 4 (True); p×q=3.24×10−12=1.8×10−6\sqrt{p \times q} = \sqrt{3.24 \times 10^{-12}} = 1.8 \times 10^{-6} (True); p−q=2.7×10−6≠2.7×10−7p - q = 2.7 \times 10^{-6} \neq 2.7 \times 10^{-7} (False); p2+q2≠p+q\sqrt{p^2 + q^2} \neq p + q (False).
Evaluating each given equation determines all correct options.

Key Concept

Decimal arithmetic and scientific notation require aligning powers of ten for addition/subtraction and proper application of radical properties.
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Topics

Decimals and Scientific Notation

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