3.2.1 Try some yourself

Answer

• (a) Estimate: 2521 = 2.521 × 10³ 3 × 10³

  •    38 = 3.8 × 10¹ 4 × 10¹.

  • So 2521 ÷ 38 = × 10³⁻¹

  • = 0.75 × 10² = 75.

  • Calculate: 2521 ÷ 38 = 66.3421 rounded to six significant figures.

• (b) Estimate: 17.85 = 1.785 × 10¹ 2 × 10¹

  •   286.3 = 2.863 × 10² 3 × 10².

  • So 17.85 × 286.3 2 × 10¹ × 3 × 10²

  • = 2 × 3 × 10¹ × 10² = 6 × 10³ = 6000.

  • Calculate: 17.85 × 286.3 = 5110.46 rounded to six significant figures.

• (c) Estimate: 1452 = 1.452 × 10³ 1.4 × 10³

  •    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 × 10³⁺³ = 0.2 × 10⁶ = 2 × 10⁻¹ × 10⁶

  • = 2 × 10⁻¹⁺⁶ = 2 × 10⁵ = 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.

Answer

Estimate: The centre is roughly 3 × 10⁴ light years away, and each light year is roughly 9 × 10¹² km, so the distance to the centre is approximately 3 × 10⁴ × 9 × 10¹² = 27 × 10¹⁶ = 2.7 × 10¹⁷.

Calculate:

  2.6 × 10⁴ × 9.46 × 10¹² = 2.4596 × 10¹⁷.