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Difficulty: MediumDecimals and Scientific Notation

Consider the expression K=0.000072×(1.5×10−4)3.6×10−11K = \frac{0.000072 \times (1.5 \times 10^{-4})}{3.6 \times 10^{-11}}. Which of the following values or expressions are equivalent to KK? Select all that apply.

  1. 3.0×1023.0 \times 10^2Answer
  2. 0.3×1030.3 \times 10^3Answer
  3. 30,000×10−230,000 \times 10^{-2}Answer
  4. D
    3.0×10−23.0 \times 10^{-2}
  5. E
    3.0×1043.0 \times 10^4

Answer

The expressions equivalent to KK are 3.0×1023.0 \times 10^2, 0.3×1030.3 \times 10^3, and 30,000×10−230,000 \times 10^{-2}.
Evaluating the expression KK gives 300300. First, rewrite 0.0000720.000072 as 7.2×10−57.2 \times 10^{-5}. Multiplying by 1.5×10−41.5 \times 10^{-4} gives (7.2×1.5)×10−9=10.8×10−9(7.2 \times 1.5) \times 10^{-9} = 10.8 \times 10^{-9}. Next, dividing by 3.6×10−113.6 \times 10^{-11} yields (10.8/3.6)×10−9−(−11)=3.0×102=300(10.8 / 3.6) \times 10^{-9 - (-11)} = 3.0 \times 10^2 = 300. Testing the choices shows that 3.0×102=3003.0 \times 10^2 = 300, 0.3×103=3000.3 \times 10^3 = 300, and 30,000×10−2=30030,000 \times 10^{-2} = 300 are all equal to 300300.

Step-by-Step Solution

1
Convert decimal numbers in the numerator to scientific notation.
0.000072=7.2×10−50.000072 = 7.2 \times 10^{-5}.
Converting all terms to powers of 10 simplifies multiplication.
2
Multiply the terms in the numerator.
(7.2×10−5)×(1.5×10−4)=(7.2×1.5)×10−5+(−4)=10.8×10−9(7.2 \times 10^{-5}) \times (1.5 \times 10^{-4}) = (7.2 \times 1.5) \times 10^{-5 + (-4)} = 10.8 \times 10^{-9}.
Coefficients multiply together and exponents add during multiplication of powers with the same base.
3
Divide the numerator by the denominator.
10.8×10−93.6×10−11=(10.83.6)×10−9−(−11)=3.0×102=300\frac{10.8 \times 10^{-9}}{3.6 \times 10^{-11}} = \left(\frac{10.8}{3.6}\right) \times 10^{-9 - (-11)} = 3.0 \times 10^2 = 300.
Dividing coefficients gives 3.03.0, and subtracting the exponent of the denominator −11-11 from −9-9 yields −9+11=2-9 + 11 = 2.
4
Verify each option against the value 300300.
3.0×102=3003.0 \times 10^2 = 300, 0.3×103=3000.3 \times 10^3 = 300, and 30,000×10−2=30030,000 \times 10^{-2} = 300 are all equivalent to 300300.
Matching each option's evaluated value ensures all correct representations are selected.

Key Concept

Simplifying numerical expressions involving decimals and scientific notation rules.
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Topics

Decimals and Scientific Notation

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