Question

Use scientific notation to estimate the number of inches in 1,225 miles. Include all calculations in your final answer. 1 inch ≈ 1.578 × 10 -^5 miles.

Answer 1

Answer:

  7.763×10^7 inches

Step-by-step explanation:

  $(1225\,mi)\dfrac{1\,in}{1.578\cdot 10^{-5}\,mi}=\dfrac{12.25}{1.578}\cdot 10^{2-(-5)}\,in \approx 7.763\cdot 10^7\,in$

When writing 1225 in scientific notation, it is convenient to choose an exponent so that the division result comes out with the right scale factor. It is easy to see that 1.2 < 1.6, so use of numbers with exactly one digit to the left of the decimal point would result in a fraction as an answer. We want the answer to have 1 digit left of the decimal point, so by scaling the operands to be 12. and 1.6, we get that result. Of course, the powers of 10 are adjusted accordingly.

This scaling process only matters if you're computing the results by hand. Any calculator can properly keep track of the required exponents.

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"Back in the day" when slide rules were the calculating tool of choice, the operands and answer to any multiplication or division problem were only shown as 3 (sometimes 4) significant digits. All of the power-of-ten scaling was done "by hand" (usually, mentally). That is, the division would be 1225/1578 ≈ 776 × some scale factor. This is where thinking of the numbers as 12.25/1.578 = 7.76 comes in handy.