3.2.1 Try some yourself
Activity 37
Use the method outlined in Example 9 to estimate each of the following, and then use your calculator to evaluate each, giving your answers to six significant figures.
(a) 2521 ÷ 38
(b) 17.85 × 286.3
(c) 1452 ÷ 0.0072
(d) 0.0053 × 0.0078 ÷ 0.6821
(e) 0.000 923 × 0.007 67.
Answer

(a) Estimate: 2521 = 2.521 × 10^{3} 3 × 10^{3}
38 = 3.8 × 10^{1} 4 × 10^{1}.
So 2521 ÷ 38 = × 10^{3−1}
= 0.75 × 10^{2} = 75.
Calculate: 2521 ÷ 38 = 66.3421 rounded to six significant figures.

(b) Estimate: 17.85 = 1.785 × 10^{1} 2 × 10^{1}
286.3 = 2.863 × 10^{2} 3 × 10^{2}.
So 17.85 × 286.3 2 × 10^{1} × 3 × 10^{2}
= 2 × 3 × 10^{1} × 10^{2} = 6 × 10^{3} = 6000.
Calculate: 17.85 × 286.3 = 5110.46 rounded to six significant figures.

(c) Estimate: 1452 = 1.452 × 10^{3} 1.4 × 10^{3}
0.0072 = 7.2 × 10^{3} 7 × 10^{−3}.
(Notice that in this case it is easier to round to 1.4 than to 1 or 1.5, anticipating the next step, since it is easier to find 1.4 ÷ 7 than 1 ÷ 7 or 1.5 ÷ 7.)
So 1452 ÷ 0.0072 = × 10^{3–−3}
= 0.2 × 10^{3+3} = 0.2 × 10^{6} = 2 × 10^{−1} × 10^{6}
= 2 × 10^{−1+6} = 2 × 10^{5} = 200 000.
Calculate: 1452 ÷ 0.0072 = 201 667 rounded to six significant figures.

(d) Estimate: 0.0053 = 5.3 × 10^{−3} 5 × 10^{−3}
0.0078 = 7.8 × 10^{−3} 8 × 10^{−3}
0.6821 = 6.821 × 10^{−1} 7 × 10^{−1}.
So
Calculate: 0.0053 × 0.0078 ÷ 0.6821
= 6.06069 × 10^{−5} = 0.000 060 6069 rounded to six significant figures.
(Notice that the calculator gives the answer in scientific notation.)
 (e)
0.000 923 × 0.007 67 = 9.23 × 10^{−4} × 7.67 × 10^{−3}
Estimate:
Calculate:
0.000 923 × 0.007 67 = 7.079 41 × 10^{−6}.
Activity 38
A light year is the distance light travels through space in one year. The centre of the Milky Way galaxy is 2.6 × 10^{4} light years away from Earth. One light year is 9.46 × 10^{12} km. How far away is the centre of the Milky Way in kilometers? Estimate your answer first and then use your calculator to do the calculation, giving your answer in scientific notation.
Answer
Estimate: The centre is roughly 3 × 10^{4} light years away, and each light year is roughly 9 × 10^{12} km, so the distance to the centre is approximately 3 × 10^{4} × 9 × 10^{12} = 27 × 10^{16} = 2.7 × 10^{17}.
Calculate:
2.6 × 10^{4} × 9.46 × 10^{12} = 2.4596 × 10^{17}.