Explanation:
a. It is so easy to double a number that no approximation is necessary.
31,000 × 200 = 3.1×10⁴ × 2×10² = 6.2×10⁶ exactly
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b. 6.2×10³ × 5.2×10⁶ ≈ 6×5×10⁹ = 3×10¹⁰ approximately
The approximation can be refined a bit by taking the ".2" into account:
6.2×10³ × 5.2×10⁶ ≈ (6×5 + .2×(6+5))×10⁹
= 32.2×10⁹ = 3.22×10¹⁰ approximately
Actual product: 3.224×10¹⁰.
For most purposes, the approximation is an adequate approximation, as it it within 10% of the actual value.
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c. 9×10⁶ -2.3×10⁴ ≈ 9×10⁶ approximately
A better approximation is to actually subtract an approximation of the smaller number:
9×10⁶ -2.3×10⁴ ≈ (9 - 0.02)×10⁶ ≈ 8.98×10⁶ approximately
The actual value uses all of the digits of the smaller number:
9×10⁶ -2.3×10⁴ ≈ (9 - 0.023)×10⁶ = 8.977×10⁶ exactly
As in part B, either approximation is adequate for most purposes, as the difference from the actual value is less than .3%.
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The accuracy required of an approximation, hence the work you expend improving accuracy, should depend on the need in the final application of the number. Often, approximations are used for budget or resource planning purposes where some "slop" is allowed or even expected.
They can also be used in engineering applications, where the error needs to be on the side of more safety (rather than less).
