شرح الصورة القياسية

Worked example

1. 1

   $7.2\times 10^4$ — The front number 7.2 is at least 1 and less than 10, so this is standard form.

2. 2

   $=7.2\times 10000$ — $10^4$ means 10 × 10 × 10 × 10, which is 10 000.

3. 3

   $=72000$ — Multiplying by 10 000 moves the point 4 places to the right: 7.2 becomes 72 000.

Worked example

1. 1

   $9400000$ — The question. It is a big number, so the power will be positive.

2. 2

   $9.4$ — Put the point after the first digit, 9. This makes the front number, which must be from 1 up to 10.

3. 3

   $9.4\to 9400000$ — Count the jumps to get back to the question: the point jumps 6 places right.

4. 4

   $9.4\times 10^6$ — 6 jumps of × 10 is $\times 10^6$. Big number, so the power is positive.

Worked example

1. 1

   $0.00048$ — The question. It is less than 1, so the power will be negative.

2. 2

   $4.8$ — Put the point after the first digit that is not 0, which is 4. This makes the front number. Zeros at the front do not count.

3. 3

   $4.8\to 0.00048$ — Count the jumps to get back to the question: the point jumps 4 places left.

4. 4

   $4.8\times 10^{-4}$ — 4 jumps of ÷ 10 is $\times 10^{-4}$. Less than 1, so the power is negative.

Worked example

1. 1

   $360\times 10^5$ — 360 is not less than 10, so this is not standard form yet.

2. 2

   $360=3.6\times 10^2$ — Write 360 in standard form on its own: the point jumps 2 places.

3. 3

   $3.6\times 10^2\times 10^5$ — Put that in place of 360. The value has not changed.

4. 4

   $3.6\times 10^7$ — Multiplying adds the powers: 2 + 5 = 7. Now 3.6 is from 1 up to 10.

Worked example

1. 1

   $3.07\times 10^5$ — The power is +5, so move the point 5 places right. A positive power means a big number.

2. 2

   $3.07\to 307$ — The first 2 jumps use the digits that are already there, 0 and 7.

3. 3

   $307\to 307000$ — 3 more jumps, with no digits left: put a 0 in each empty place. That is 5 jumps in all.

Worked example

1. 1

   $6.1\times 10^{-3}$ — The power is −3, so move the point 3 places left. A negative power means a number less than 1.

2. 2

   $6.1\to 0.61$ — Jump 1: the point moves to the front of the 6.

3. 3

   $0.61\to 0.0061$ — Jumps 2 and 3: put a 0 in each empty place in front of the 6.

Worked example

1. 1

   $(3\times 10^5)\times (6\times 10^4)$ — The question. You can multiply in any order, so the front numbers and the powers can be done separately.

2. 2

   $3\times 6=18$ — Multiply the front numbers as normal.

3. 3

   $10^5\times 10^4=10^9$ — Multiplying powers of 10 adds the powers: 5 + 4 = 9.

4. 4

   $18\times 10^9$ — Put the two parts together. Not finished: 18 is not less than 10.

5. 5

   $1.8\times 10^{10}$ — 18 is 1.8 × 10. That extra 10 joins the power, so 9 + 1 = 10.

Worked example

1. 1

   $(1.2\times 10^9)÷(4\times 10^3)$ — The question. Dividing works the same way: front numbers on their own, powers on their own.

2. 2

   $1.2÷4=0.3$ — Divide the front numbers as normal.

3. 3

   $10^9÷10^3=10^6$ — Dividing powers of 10 subtracts the powers: 9 − 3 = 6.

4. 4

   $0.3\times 10^6$ — Put the two parts together. Not finished: 0.3 is less than 1.

5. 5

   $3\times 10^5$ — 0.3 is 3 ÷ 10. Dividing by 10 takes 1 off the power, so 6 − 1 = 5.

Worked example

1. 1

   $(3\times 10^5)÷(6\times 10^{-3})$ — The question. The second power is negative.

2. 2

   $3÷6=0.5$ — Divide the front numbers as normal.

3. 3

   $10^5÷10^{-3}=10^{5-(-3)}$ — Dividing subtracts the powers. Keep the −3 in brackets so its minus is not lost.

4. 4

   $5-(-3)=5+3=8$ — Taking away a negative is the same as adding: the two minuses make a plus.

5. 5

   $0.5\times 10^8$ — Put the two parts together. Not finished: 0.5 is less than 1.

6. 6

   $5\times 10^7$ — 0.5 is 5 ÷ 10, so take 1 off the power: 8 − 1 = 7.

Worked example

1. 1

   $(3\times 10^{-4}{)}^2$ — Squared means times itself: $(3\times 10^{-4})\times (3\times 10^{-4})$.

2. 2

   $3^2=9$ — Square the front number: 3 × 3 = 9.

3. 3

   $(10^{-4}{)}^2=10^{-8}$ — Squaring a power doubles it: −4 × 2 = −8. It is the same as −4 + (−4).

4. 4

   $9\times 10^{-8}$ — 9 is from 1 up to 10, so this is already standard form.

Worked example

1. 1

   $5.2\times 10^6+3.4\times 10^5$ — Find the total. The powers are different (6 and 5), so the front numbers cannot simply be added.

2. 2

   $5.2\times 10^6=5200000$ — Write the first as an ordinary number: the point moves 6 places right.

3. 3

   $3.4\times 10^5=340000$ — Write the second as an ordinary number: the point moves 5 places right.

4. 4

   $5200000+340000=5540000$ — Add as normal, with the digits lined up by place value.

5. 5

   $=5.54\times 10^6$ — Back to standard form: point after the 5, and it jumps 6 places.

Worked example

1. 1

   $7\times 10^4-8\times 10^3$ — Find the difference. Again the powers are different.

2. 2

   $7\times 10^4=70000$ — Write the first as an ordinary number: the point moves 4 places right.

3. 3

   $8\times 10^3=8000$ — Write the second as an ordinary number: the point moves 3 places right.

4. 4

   $70000-8000=62000$ — Subtract as normal.

5. 5

   $=6.2\times 10^4$ — Back to standard form: point after the 6, and it jumps 4 places.

Worked example

1. 1

   $4.6\times 10^{120}+7\times 10^{119}$ — Find the total. The powers are 120 and 119, so make them the same first.

2. 2

   $7\times 10^{119}=0.7\times 10^{120}$ — Rewrite the smaller one. Dividing the front number by 10 and adding 1 to the power keeps the value the same.

3. 3

   $4.6\times 10^{120}+0.7\times 10^{120}$ — Now both have $10^{120}$.

4. 4

   $(4.6+0.7)\times 10^{120}$ — Same power, so add the front numbers. The $10^{120}$ stays as it is.

5. 5

   $=5.3\times 10^{120}$ — 4.6 + 0.7 = 5.3. It is from 1 up to 10, so this is the answer.

Worked example

1. 1

   $3\times 10^{80}-5\times 10^{78}$ — Find the difference. The powers are 2 apart.

2. 2

   $5\times 10^{78}=0.05\times 10^{80}$ — Move the point 2 places left in 5 and add 2 to the power. The value stays the same.

3. 3

   $(3-0.05)\times 10^{80}$ — Same power, so subtract the front numbers.

4. 4

   $=2.95\times 10^{80}$ — 3 − 0.05 = 2.95. It is from 1 up to 10, so this is the answer.

Worked example

1. 1

   $\frac{4.8\times 10^6}{1.2\times 10^4}$ — “How many times” means divide the bigger number by the smaller one.

2. 2

   $4.8÷1.2=4$ — Divide the front numbers.

3. 3

   $10^6÷10^4=10^2$ — Dividing subtracts the powers: 6 − 4 = 2.

4. 4

   $4\times 10^2=400$ — Put the parts together. The city has 400 times as many people.

Worked example

1. 1

   $n=\frac{9\times 10^8}{2.5\times 10^3}$ — n is multiplied by $2.5\times 10^3$, so divide both sides by it. Division undoes multiplication.

2. 2

   $9÷2.5=3.6$ — Divide the front numbers.

3. 3

   $10^8÷10^3=10^5$ — Dividing subtracts the powers: 8 − 3 = 5.

4. 4

   $n=3.6\times 10^5$ — 3.6 is from 1 up to 10, so this is standard form. Check: 2.5 × 3.6 = 9 and 3 + 5 = 8.

Worked example

1. 1

   $\frac{5.3\times 10^4+8\times 10^3}{2.4\times 10^{-2}}$ — The question. Give the answer in standard form, correct to 3 significant figures.

2. 2

   $5.3\times 10^4+8\times 10^3=61000$ — Work out the top first, and write it down. A correct line like this can earn a mark even if a later step slips.

3. 3

   $61000÷(2.4\times 10^{-2})=2541666.6\dots$ — Now divide by the bottom. The brackets make the calculator divide by all of it.

4. 4

   $2.541666\dots \times 10^6$ — Point after the 2: it jumps 6 places, so the power is 6.

5. 5

   $2.54\times 10^6$ — Round the front number to 3 significant figures: 2.541… becomes 2.54. The power does not change.

Worked example

1. 1

   $0.5\times 10^{-2}=5\times 10^{-3}$ — First fix any number that is not in standard form. 0.5 is less than 1, so make it 5 and take 1 off the power.

2. 2

   $10^{-3}<10^{-2}<10^1$ — Compare the powers. −3 is smaller than −2, so the $10^{-3}$ numbers are the smallest.

3. 3

   $5\times 10^{-3}<6.2\times 10^{-3}$ — Two numbers share $10^{-3}$, so compare their front numbers: 5 is less than 6.2.

4. 4

   $0.5\times 10^{-2},6.2\times 10^{-3},1.9\times 10^{-2},4\times 10^1$ — The order, smallest first. Write each number the way the question gave it.