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Difficulty: MediumExponents, Roots, and Scientific Notation

Arrange the following four mathematical expressions in order from least to greatest value.

  1. 19×10−40.5\frac{9 \times 10^{-4}}{0.5}
  2. 2(5×10−2)2\left(5 \times 10^{-2}\right)^2
  3. 33.2×10−33.2 \times 10^{-3}
  4. 41.6×10−5\sqrt{1.6 \times 10^{-5}}

Answer

The correct order from least to greatest value is 9×10−40.5\frac{9 \times 10^{-4}}{0.5}, (5×10−2)2\left(5 \times 10^{-2}\right)^2, 3.2×10−33.2 \times 10^{-3}, and 1.6×10−5\sqrt{1.6 \times 10^{-5}}.
Converting each expression to standard scientific notation with base power 10−310^{-3} allows for direct comparison of coefficients: 9×10−40.5=1.8×10−3\frac{9 \times 10^{-4}}{0.5} = 1.8 \times 10^{-3}, (5×10−2)2=2.5×10−3\left(5 \times 10^{-2}\right)^2 = 2.5 \times 10^{-3}, 3.2×10−33.2 \times 10^{-3} is already in standard form, and 1.6×10−5=16×10−6=4.0×10−3\sqrt{1.6 \times 10^{-5}} = \sqrt{16 \times 10^{-6}} = 4.0 \times 10^{-3}. Comparing coefficients 1.8<2.5<3.2<4.01.8 < 2.5 < 3.2 < 4.0 confirms the order from least to greatest.

Step-by-Step Solution

1
Evaluate the expression 9×10−40.5\frac{9 \times 10^{-4}}{0.5}
1.8×10−31.8 \times 10^{-3} (or 0.00180.0018)
Dividing the coefficient 99 by 0.50.5 gives 1818. Then convert 18×10−418 \times 10^{-4} to standard scientific notation 1.8×10−31.8 \times 10^{-3}.
2
Evaluate the expression (5×10−2)2\left(5 \times 10^{-2}\right)^2
2.5×10−32.5 \times 10^{-3} (or 0.00250.0025)
Apply the power of a product rule: 52×(10−2)2=25×10−4=2.5×10−35^2 \times (10^{-2})^2 = 25 \times 10^{-4} = 2.5 \times 10^{-3}.
3
Evaluate the expression 1.6×10−5\sqrt{1.6 \times 10^{-5}}
4.0×10−34.0 \times 10^{-3} (or 0.00400.0040)
Rewrite 1.6×10−51.6 \times 10^{-5} as 16×10−616 \times 10^{-6} so the exponent of 1010 is even. Taking the square root gives 16×10−6=4×10−3\sqrt{16} \times \sqrt{10^{-6}} = 4 \times 10^{-3}.
4
Compare all expressions in standard scientific notation with power 10−310^{-3}
1.8×10−3<2.5×10−3<3.2×10−3<4.0×10−31.8 \times 10^{-3} < 2.5 \times 10^{-3} < 3.2 \times 10^{-3} < 4.0 \times 10^{-3}
Since all numbers share the same power of ten (10−310^{-3}), compare their coefficients: 1.8<2.5<3.2<4.01.8 < 2.5 < 3.2 < 4.0.

Key Concept

Comparing quantities in scientific notation by simplifying exponent and radical operations to standard forms.
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Topics

Exponents, Roots, and Scientific Notation

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