Calculating in Standard Form

Fronts multiply; powers of ten add.

Multiplying

To multiply (3 × 10⁵) × (2 × 10³), remember that numbers being multiplied can be multiplied in any order. So put the numbers in front together and the powers of ten together: (3 × 2) × (10⁵ × 10³).

The numbers in front multiply: 3 × 2 = 6. The powers of ten follow the law of indices for multiplying: 10⁵ × 10³ = 10^(5 + 3) = 10⁸, because five copies of 10 beside three more copies make eight copies. So (3 × 10⁵) × (2 × 10³) = 6 × 10⁸.

10·10·10·10·1010·10·1010⁵ × 10³ = 10⁸

Five copies of 10 beside three more copies make eight copies of 10, so the indices add: 5 + 3 = 8.

Dividing

Dividing works the same way, with each operation turned around. To find (8 × 10⁷) / (2 × 10³), divide the numbers in front, 8 ÷ 2 = 4, and divide the powers of ten by subtracting the indices: 10⁷ / 10³ = 10^(7 − 3) = 10⁴. So the answer is 4 × 10⁴.

Subtracting the indices is canceling copies. 10⁷ is seven copies of 10, and dividing by 10³ cancels three of them, which leaves four.

10·10·10·10·10·10·1010⁷ ÷ 10³ = 10⁴

Seven copies of 10 with three canceled leave four copies: 10⁷ / 10³ = 10⁴.

Adding needs equal powers

Adding is different. In 3 × 10⁵ + 4 × 10⁴, the 3 counts lots of 10⁵ and the 4 counts lots of 10⁴. They count different things, so 3 + 4 would be as wrong as adding 3 meters to 4 centimeters and calling it 7.

First rewrite one number so that both use the same power of ten. 10⁵ is ten times 10⁴, so 3 lots of 10⁵ are 30 lots of 10⁴: 3 × 10⁵ = 30 × 10⁴. Now both count lots of 10⁴, and they add: 30 × 10⁴ + 4 × 10⁴ = 34 × 10⁴.

34 is not between 1 and 10, so write it as 3.4 × 10 and move that ten into the power: 34 × 10⁴ = 3.4 × 10⁵. The power of ten stays the same while adding; only the numbers in front are added.

3 × 10⁵4 × 10⁴

Every part is 10⁴. The first bar, 3 × 10⁵, is 30 parts; the second, 4 × 10⁴, is 4 parts. Together they are 34 parts, 34 × 10⁴.

When the front grows past 10

Multiplying the numbers in front can give 10 or more. (6 × 10⁵) × (5 × 10³) = 30 × 10⁸, and 30 is too big for standard form.

Take a factor of 10 out of the front and put it into the power of ten: 30 = 3 × 10, so 30 × 10⁸ = 3 × 10 × 10⁸ = 3 × 10⁹. The front gets 10 times smaller, so the power of ten gets one higher.

1000010001001010.10.010.0013030÷ 1000÷ 100÷ 10× 1× 10× 100× 1000

× 1 moves no digit, so the value is unchanged

Turn 30 into 3

Choose ÷ 10: each digit of 30 moves one place to the right, and 30 becomes 3. To keep the value, the power of ten must go up by one, from 10⁸ to 10⁹.

Two common mistakes

When multiplying, the indices add; they do not multiply. (2 × 10⁴) × (3 × 10²) = 6 × 10⁶, not 6 × 10⁸, because four copies of 10 beside two more make six.

When adding, the indices do not add at all. 2 × 10⁶ + 5 × 10⁶ = 7 × 10⁶, not 7 × 10¹²: the power stays 10⁶, and only the numbers in front are added.

Worked example: The Distances of Two Planets from the Sun in Standard Form

Question The Earth is about 1.5 × 108 km from the Sun and Neptune is about 4.5 × 109 km from the Sun. (a) How many times as far from the Sun as the Earth is Neptune? (b) At one time the Earth and Neptune are in a straight line with the Sun, on opposite sides of it. Find the distance between the two planets then, in standard form.

  1. 1.The question "how many times as far" is a division: (4.5 × 109) ÷ (1.5 × 108).

    Neptune 4.5 × 109km, Earth 1.5 × 108km(4.5 × 109) divided by (1.5 × 108)
    Neptune 4.5 × 109km, Earth 1.5 × 108km(4.5 × 109) divided by (1.5 × 108)
    "How many times as far" is a division: (4.5 × 109) ÷ (1.5 × 108).

  2. 2.Divide the numbers and the powers of ten separately. 4.5 ÷ 1.5 = 3, and 109 ÷ 108 = 109 − 8 = 101.

    Neptune 4.5 × 109km, Earth 1.5 × 108km4.5 / 1.5 = 3109/ 108= 109 − 8= 101
    Neptune 4.5 × 109km, Earth 1.5 × 108km4.5 / 1.5 = 3109/ 108= 109 − 8= 101
    Divide the numbers and the powers of ten separately: 4.5 ÷ 1.5 = 3 and 109 ÷ 108 = 101.

  3. 3.(a) 3 × 101 = 30, so Neptune is 30 times as far from the Sun as the Earth is.

    Neptune 4.5 × 109km, Earth 1.5 × 108km4.5 / 1.5 = 3 and 109/ 108= 1013 × 101= 30 times as far
    Neptune 4.5 × 109km, Earth 1.5 × 108km4.5 / 1.5 = 3 and 109/ 108= 1013 × 101= 30 times as far
    (a) 3 × 101 = 30, so Neptune is 30 times as far from the Sun.

  4. 4.With the Sun between them, the distance between the planets is the sum of the two distances. To add, the powers of ten must be equal. Write the Earth's distance with 109: the power of ten becomes 10 times as large, so the number in front becomes 10 times as small, 1.5 × 108 = 0.15 × 109.

    Neptune 4.5 × 109km, Earth 1.5 × 108km109108107Neptune450Earth0151.5 × 108= 0.15 × 109
    Neptune 4.5 × 109km, Earth 1.5 × 108km109108107Neptune450Earth0151.5 × 108= 0.15 × 109
    To add, the powers of ten must be equal: 1.5 × 108 = 0.15 × 109.

  5. 5.(b) 4.5 × 109 + 0.15 × 109 = (4.5 + 0.15) × 109 = 4.65 × 109 km. Since 4.65 lies between 1 and 10, this is already in standard form.

    Neptune 4.5 × 109km, Earth 1.5 × 108km109108107Neptune450Earth015sum465(4.5 + 0.15) × 109= 4.65 × 109km
    Neptune 4.5 × 109km, Earth 1.5 × 108km109108107Neptune450Earth015sum465(4.5 + 0.15) × 109= 4.65 × 109km
    (b) (4.5 + 0.15) × 109 = 4.65 × 109 km.

Answer: (a) 30 times as far; (b) 4.65 × 109 km

Common mistakes

  • Adding the numbers in front without making the powers equal, (4.5 + 1.5) × 109 = 6 × 109. The 1.5 counts lots of 108 and the 4.5 counts lots of 109, so the Earth's distance must first be written as 0.15 × 109.
  • Adding the indices in part (b) to get 1017. Indices are added when powers are multiplied. In an addition with equal powers of ten, the power stays as 109 and only the numbers in front are added.

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