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Elite Mathematics
IGCSE · 4MA1 · Topic

Powers, Roots and Standard Form: past paper questions & worked solutions

Every Powers, Roots and Standard Form question from the 4MA1 papers in our bank, 2020–2026, plus original Elite practice questions, sorted from easiest to hardest. Try each one first, then open the worked solution.

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35questions
112marks in total
56%of papers (30 of 54) include it
8/24/3easy / medium / hard

Questions, easiest first

Level is estimated from marks: 1–2 easy, 3–4 medium, 5+ hard.

1. Nov 2025 4WM2H · Q13

Easy2 marks
Nov 2025 4WM2H Question 13 (2 marks): Powers, Roots and Standard Form – 13 Work out 9 1080 3 Give your answer in standard form. (Total for Question 13 is 2 marks)
Show worked solution
  1. Use index laws

    (9×10<sup>80</sup>)<sup>3</sup> = 9<sup>3</sup> × (10<sup>80</sup>)<sup>3</sup>

  2. Solve inequality

    9<sup>3</sup> = 729 (10<sup>80</sup>)<sup>3</sup> = 10<sup>80 × 3</sup> = 10<sup>240</sup>

  3. Convert standard form

    729 × 10<sup>240</sup> = 7.29 × 10<sup>2</sup> × 10<sup>240</sup> = 7.29 × 10<sup>242</sup>

Final answer

7.29 × 10<sup>242</sup>

2. Jun 2025 P1HR · Q14

Easy2 marks
Jun 2025 P1HR Question 14 (2 marks): Powers, Roots and Standard Form – 14 Work out 6 7 10 3 10 135 134 . Give your answer in standard form. (Total for Question 14 is 2 marks)
Show worked solution
  1. Convert standard form
    \[ 6.7\times 10^{135}+3\times 10^{134} \]
  2. Write both terms using

    Write both terms using \(10^{135}\):

    \[ 3\times 10^{134}=0.3\times 10^{135} \]
  3. Convert standard form

    So

    \[ 6.7\times 10^{135}+0.3\times 10^{135}=7.0\times 10^{135} \]
    \[ =7\times 10^{135} \]
Final answer

\(\displaystyle 7\times 10^{135}\)

3. Jun 2023 P2H · Q6

Easy2 marks
Jun 2023 P2H Question 6 (2 marks): Powers, Roots and Standard Form – 6 (a) Write 76 000 000 in standard form. (1) (b) Write 5.4 × 10-4 as an ordinary number. (1) (Total for Question 6 is 2 marks)
Show worked solution
  1. Convert standard form
    \[ 76000000=7.6\times 10^7 \]
    \[ 5.4\times 10^{-4}=0.00054 \]
Final answer

(a) \(7.6\times 10^7\), (b) \(0.00054\)

4. Jun 2022 P1H · Q8

Easy2 marks
Jun 2022 P1H Question 8 (2 marks): Powers, Roots and Standard Form – 8 (a) Write 0.000 089 in standard form. (1) (b) Write 8.34 × 104 as an ordinary number. (1) (Total for Question 8 is 2 marks)
Show worked solution
  1. Convert standard form
    \[ 0.000089=8.9\times 10^{-5} \]
    \[ 8.34\times 10^4=83400 \]
Final answer

(a) \(8.9\times 10^{-5}\), (b) \(83400\)

5. Jan 2022 P1H · Q8

Easy2 marks
Jan 2022 P1H Question 8 (2 marks): Powers, Roots and Standard Form – 8 a = 4.2 × 10-24 b = 3 × 10145 Work out the value of a × b Give your answer in standard form. (Total for Question 8 is 2 marks)
Show worked solution
  1. Convert standard form
    \[ a=4.2\times 10^{-24} \]
    \[ b=3\times 10^{145} \]
  2. Convert standard form

    So

    \[ ab=(4.2\times 3)\times 10^{-24+145} \]
    \[ ab=12.6\times 10^{121} \]
  3. Write this in standard form

    Write this in standard form:

    \[ 12.6\times 10^{121}=1.26\times 10^{122} \]
Final answer

\(\displaystyle 1.26\times 10^{122}\)

6. Jan 2021 P1H · Q6

Easy2 marks
Jan 2021 P1H Question 6 (2 marks): Powers, Roots and Standard Form – 6 (a) Write 2 840 000 000 in standard form. (1) (b) Write 2.5 × 10 -4 as an ordinary number. (1) (Total for Question 6 is 2 marks)
Show worked solution
  1. Convert standard form
    \[ 2840000000=2.84\times 10^9 \]
  2. Convert standard form

    Also,

    \[ 2.5\times 10^{-4}=0.00025 \]
Final answer

(a) \(2.84\times 10^9\), (b) \(0.00025\)

7. Specimen 4WM2H · Q9

Easy2 marks
Specimen 4WM2H Question 9 (2 marks): Powers, Roots and Standard Form – 9 (a) Write 0.000089 in standard form. (b) Write 8.34 x 10^4 as an ordinary number. (Total for Question 9 is 2 marks)
Show worked solution
  1. Convert standard form
    \[ 0.000089=8.9\times 10^{-5} \]
  2. Convert standard form

    Also

    \[ 8.34\times 10^4=83400 \]
Final answer

\(8.9\times10^{-5}\), \(83400\).

8. Elite Practice · Q34

Easy2 marks
Elite Practice Question 34 (2 marks): Powers, Roots and Standard Form – 34 Write (3^5 x 3^-2) / 3^4 as a single power of 3. (Total for Question 34 is 2 marks)
Show worked solution
  1. Multiply: add the powers

    \(3^5\times3^{-2}=3^{3}\)

  2. Divide: subtract the powers
    \[ \frac{3^3}{3^4}=3^{-1} \]
Final answer

\(3^{-1}\)

9. Jun 2026 P2HR · Q7

Medium3 marks
Jun 2026 P2HR Question 7 (3 marks): Powers, Roots and Standard Form – 7 (a) Write 4 × 10 –5 as an ordinary number. (1) (b) Work out (6 × 10 158) × (2 × 10 72) Give your answer in standard form. (2) (Total for Question 7 is 3 marks)
Show worked solution
  1. Part (a) | Step 1
    \[4\times10^{-5}=\frac4{100000}=0.00004.\]

    Move the decimal point five places left.

  2. Part (b) | Step 1
    \[(6\times10^{158})(2\times10^{72})=12\times10^{230}=1.2\times10^{231}.\]

    Multiply the coefficients and add the exponents; normalize the coefficient into [1,10).

Final answer

(a) \(0.00004\).

(b) \(1.2\times10^{231}\).

10. Jun 2026 P1H · Q9

Medium3 marks
Jun 2026 P1H Question 9 (3 marks): Powers, Roots and Standard Form – 9 (a) Write 0.0651 in standard form. (1) (b) Work out 4 10 8 10 56 74 � � �� � � � Give your answer in standard form. (2) (Total for Question 9 is 3 marks)
Show worked solution
  1. Part (a) | Step 1
    \[0.0651=6.51\times10^{-2}.\]

    The coefficient must be at least 1 and less than 10.

  2. Part (b) | Step 1
    \[(4\times10^{56})(8\times10^{74})=32\times10^{130}=3.2\times10^{131}.\]

    Multiply the coefficients, add the exponents, and adjust the coefficient to standard form.

Final answer

(a) \(6.51\times10^{-2}\)

(b) \(3.2\times10^{131}\)

11. Nov 2025 P2H · Q8

Medium3 marks
Nov 2025 P2H Question 8 (3 marks): Powers, Roots and Standard Form – 8 (a) Write 5.76 × 10 4 as an ordinary number. (1) (b) Work out 5 3 2 3 10 8 10 4 10 Give your answer in standard form. (2) (Total for Question 8 is 3 marks)
Show worked solution
  1. Convert standard form
    \[ 5.76\times 10^4=57600 \]
  2. Convert standard form

    For part (b), simplify the numerator:

    \[ 3\times 10^5+8\times 10^3=300000+8000=308000 \]
  3. Convert standard form

    Then

    \[ \frac{308000}{4\times 10^{-2}}=\frac{308000}{0.04}=7700000 \]
  4. In standard form

    In standard form:

    \[ 7700000=7.7\times 10^6 \]
Final answer

(a) \(57600\), (b) \(7.7\times 10^6\)

12. Nov 2025 4WM2H · Q9

Medium3 marks
Nov 2025 4WM2H Question 9 (3 marks): Powers, Roots and Standard Form – 9 (a) Write 5.76 × 10 4 as an ordinary number. (1) (b) Work out 5 3 2 3 10 8 10 4 10 Give your answer in standard form. (2) (Total for Question 9 is 3 marks)
Show worked solution
  1. Move the decimal point 4

    Move the decimal point 4 places to the right: 5.76 × 10<sup>4</sup> = 57600

  2. Answer

    57600

  3. Expand brackets

    3 × 10<sup>5</sup> = 300, 000 8 × 10<sup>3</sup> = 8, 000 Numerator = 300, 000 + 8, 000 = 308, 000 = 3.08 × 10<sup>5</sup>

  4. Convert standard form

    $$\\frac{3.08 \\times 10^5}{4 \\times 10^{-2}} = \\left(\\frac{3.08}{4}\\right) \\times 10^{5 - (-2)}$$  = 0.77 × 10<sup>7</sup>

    **Step 3: Convert to strict standard form (1 ≤ *A* \< 10).** 0.77 × 10<sup>7</sup> = 7.7 × 10<sup>6</sup>

Final answer

7.7 × 10<sup>6</sup>

13. May 2025 4WM2HR · Q6

Medium3 marks
May 2025 4WM2HR Question 6 (3 marks): Powers, Roots and Standard Form – 6 (a) Write 82 000 000 in standard form. (1) (b) Write 5 8 10 6 . as an ordinary number. (1) (c) Write 160 10300 × in standard form. (1) (Total for Question 6 is 3 marks)
Show worked solution
  1. Required

    \(a\) Write 82 000 000 in standard form. (b) Write 5.8 × 10<sup>−6</sup> as an ordinary number. (c) Write 160 × 10<sup>300</sup> in standard form.

  2. Part (a)

    82 000 000 = 8.2 × 10<sup>7</sup>

  3. Answer

    8.2 × 10<sup>7</sup>

  4. Part (b)

    Move the decimal 6 places **left**: 5.8 × 10<sup>−6</sup> = 0.0000058

  5. Answer

    0.0000058

  6. Part (c)

    160 × 10<sup>300</sup> = 1.6 × 10<sup>1</sup> × 10<sup>300</sup> = 1.6 × 10<sup>301</sup>

Final answer

1.6 × 10<sup>301</sup>

14. May 2025 4WM2H · Q12

Medium3 marks
May 2025 4WM2H Question 12 (3 marks): Powers, Roots and Standard Form – 12 (a) Write 8.6 × 10 −5 as an ordinary number. (1) (b) Work out Give your answer in standard form. (2) (Total for Question 12 is 3 marks)
Show worked solution
  1. (a)
    \[ 8.6\times 10^{-5}=0.000086 \]
  2. (b)
    \[ (3\times 10^{43})\div(6\times 10^{37}) \]
    \[ =0.5\times 10^6=5\times 10^5 \]
Final answer

\(0.000086\), \(5\times10^5\).

15. May 2025 4WM2HR · Q13

Medium3 marks
May 2025 4WM2HR Question 13 (3 marks): Powers, Roots and Standard Form – 13 The distance from Earth to Andromeda Galaxy is 1 6 1011 . × astronomical units. • 1 astronomical unit 1 5 108 . km • 1 light year 9 5 1012 . km Work out the distance, in light years, from Earth to Andromeda Galaxy. Gi
Show worked solution
  1. Required

    Distance to Andromeda: 1.6 × 10<sup>11</sup> AU. Convert to light years (2 s.f., standard form).

  2. Convert AU km 1 6

    Convert AU → km: 1.6 × 10<sup>11</sup> × 1.5 × 10<sup>8</sup> = (1.6×1.5) × 10<sup>19</sup> = 2.4 × 10<sup>19</sup> km

  3. Convert km light years

    Convert km → light years: $$\\frac{2.4\\times10^{19}}{9.5\\times10^{12}} = \\frac{2.4}{9.5}\\times10^{7} = 0.2526\\ldots\\times10^{7}$$

  4. Convert standard form

    Write in standard form to 2 s.f.:  = 2.526… × 10<sup>6</sup> ≈ 2.5 × 10<sup>6</sup>

Final answer

Distance  = 2.5 × 10<sup>6</sup> light years

16. Nov 2024 P1H · Q9

Medium3 marks
Nov 2024 P1H Question 9 (3 marks): Powers, Roots and Standard Form – 9 (a) Write 8 4 10 5 . as an ordinary number. (1) (b) Work out 6 5 10 8 10 40 185 . Give your answer in standard form. (2) (Total for Question 9 is 3 marks)
Show worked solution
  1. Convert standard form
    \[ 8.4\times 10^{-5}=0.000084 \]
  2. Convert standard form

    For part (b),

    \[ (6.5\times 10^{-40})(8\times 10^{185}) =52\times 10^{145} \]
  3. Write this in standard form

    Write this in standard form:

    \[ 52\times 10^{145}=5.2\times 10^{146} \]
Final answer

(a) \(0.000084\), (b) \(5.2\times 10^{146}\)

17. May 2024 P1H · Q9

Medium3 marks
May 2024 P1H Question 9 (3 marks): Powers, Roots and Standard Form – 9 The table gives the amount of rice produced by each of two countries in 2020 Country Amount of rice (tonnes) Indonesia 3.5 × 10 7 Argentina 8.2 × 10 5 (a) Write 3.5 × 10 7 as an ordinary number. (1) In 2020, Japan pr
Show worked solution
  1. Convert standard form
    \[ 3.5\times 10^7=35000000 \]
  2. Argentina produced
    \[ 8.2\times 10^5=820000 \]
  3. tonnes of rice

    tonnes of rice.

  4. Japan produced more tonnes than

    Japan produced \(6780000\) more tonnes than Argentina, so

    \[ 820000+6780000=7600000 \]
  5. Convert standard form

    In standard form,

    \[ 7600000=7.6\times 10^6 \]
Final answer

(a) \(35000000\), (b) \(7.6\times 10^6\) tonnes

18. May 2023 P1HR · Q8

Medium3 marks
May 2023 P1HR Question 8 (3 marks): Powers, Roots and Standard Form – 8 (a) Write 5.6 × 10-3 as an ordinary number. (1) (b) Work out 6 10 2 1 10 9 10 3 4 5 . Give your answer in standard form. (2) (Total for Question 8 is 3 marks)
Show worked solution
  1. Convert standard form
    \[ 5.6\times 10^{-3}=0.0056 \]
  2. Convert standard form

    For part (b), simplify the denominator:

    \[ 2.1\times 10^{-4}+9\times 10^{-5} =0.00021+0.00009 \]
    \[ =0.00030=3\times 10^{-4} \]
  3. Convert standard form

    Then

    \[ \frac{6\times 10^3}{3\times 10^{-4}} =2\times 10^{3-(-4)} \]
    \[ =2\times 10^7 \]
Final answer

(a) \(0.0056\), (b) \(2\times 10^7\)

19. Jan 2022 P2H · Q12

Medium3 marks
Jan 2022 P2H Question 12 (3 marks): Powers, Roots and Standard Form – 12 a = 6 × 1040 Work out the value of a3 Give your answer in standard form. (Total for Question 12 is 3 marks)
Show worked solution
  1. Convert standard form
    \[ a=6\times 10^{40} \]
    \[ a^3=(6\times 10^{40})^3 \]
    \[ a^3=6^3\times 10^{120} \]
    \[ a^3=216\times 10^{120} \]
  2. Write this in standard form

    Write this in standard form:

    \[ 216\times 10^{120}=2.16\times 10^{122} \]
Final answer

\(\displaystyle 2.16\times 10^{122}\)

20. Jan 2021 P2HR · Q8

Medium3 marks
Jan 2021 P2HR Question 8 (3 marks): Powers, Roots and Standard Form – 8 The table gives the length of the coastline, in kilometres, of each of five oceans. Ocean Length of coastline (km) Arctic 4.539 × 10 4 Atlantic 1.119 × 10 5 Pacific 1.357 × 10 5 Indian 6.653 × 10 4 Southern 1.797 × 10
Show worked solution
  1. The greatest coastline is the

    The greatest coastline is the largest number in standard form.

    \[ 1.357\times 10^5 \]
  2. Convert standard form

    is the largest, so the answer is the Pacific Ocean.

  3. Convert standard form

    For part (b),

    \[ 1.119\times 10^5=111900 \]
    \[ 1.797\times 10^4=17970 \]
  4. Difference

    Difference:

    \[ 111900-17970=93930 \]
    \[ 93930=9.393\times 10^4 \]
Final answer

(a) Pacific Ocean, (b) \(9.393\times 10^4\) km

21. Jan 2020 P1H · Q6

Medium3 marks
Jan 2020 P1H Question 6 (3 marks): Powers, Roots and Standard Form – 6 (a) Write 7.8 × 10-4 as an ordinary number. (1) (b) Work out 5 6 10 7 10 2 8 10 4 3 3 . . × + × × - Give your answer in standard form. (2) (Total for Question 6 is 3 marks)
Show worked solution
  1. Convert standard form
    \[ 7.8\times 10^{-4}=0.00078 \]
  2. Convert standard form

    For part (b),

    \[ 5.6\times 10^4+7\times 10^3=56000+7000=63000 \]
  3. Convert standard form

    Then

    \[ \frac{63000}{2.8\times 10^{-3}}=\frac{63000}{0.0028} \]
    \[ =22500000=2.25\times 10^7 \]
Final answer

(a) \(0.00078\), (b) \(2.25\times 10^7\)

22. Jan 2020 P2HR · Q10

Medium3 marks
Jan 2020 P2HR Question 10 (3 marks): Powers, Roots and Standard Form – 10 The table shows information about the surface area of each of the world’s oceans. Ocean Surface area in square kilometres Pacific 1.56 × 108 Indian 6.86 × 107 Southern 2.03 × 107 Arctic 1.41 × 107 Atlantic 1.06 × 108
Show worked solution
  1. Atlantic surface area

    Atlantic surface area:

    \[ 1.06\times 10^8=106000000 \]
  2. Indian surface area

    Indian surface area:

    \[ 6.86\times 10^7=68600000 \]
  3. Difference

    Difference:

    \[ 106000000-68600000=37400000 \]
    \[ 37400000=3.74\times 10^7 \]
  4. Convert standard form

    For part (b),

    \[ k=\frac{1.56\times 10^8}{1.41\times 10^7} \]
    \[ k=11.063\ldots \]
  5. Convert standard form

    To the nearest whole number,

    \[ k=11 \]
Final answer

(a) \(3.74\times 10^7\), (b) \(11\)

23. Elite Practice · Q33

Medium3 marks
Elite Practice Question 33 (3 marks): Powers, Roots and Standard Form – 33 Find the value of (a) 5^0 (b) 8^(2/3) (c) 16^(-1/2). (Total for Question 33 is 3 marks)
Show worked solution
  1. (a) Any non-zero number to the power 0

    \(5^0=1\)

  2. (b) Root first, then power
    \[ 8^{2/3}=\left(\sqrt[3]{8}\right)^2=2^2=4 \]
  3. (c) Negative power means reciprocal
    \[ 16^{-1/2}=\frac{1}{\sqrt{16}}=\frac14 \]
Final answer

(a) \(1\) (b) \(4\) (c) \(\frac14\)

24. Elite Practice · Q35

Medium3 marks
Elite Practice Question 35 (3 marks): Powers, Roots and Standard Form – 35 2^n = 8^4 ÷ 4^3. Find the value of n. (Total for Question 35 is 3 marks)
Show worked solution
  1. Write everything as a power of 2

    \(8=2^3\), so \(8^4=2^{12}\). \(4=2^2\), so \(4^3=2^{6}\).

  2. Divide
    \[ \frac{2^{12}}{2^{6}}=2^{6} \;\Rightarrow\; n=6 \]
Final answer

\(n=6\)

25. Jun 2025 P1H · Q8

Medium4 marks
Jun 2025 P1H Question 8 (4 marks): Powers, Roots and Standard Form – 8 (a) Write 520 million in standard form. (1) (b) Write 8.79 × 10 -5 as an ordinary number. (1) (c) Work out (5 × 10 42) × (7 × 10 -180) Give your answer in standard form. (2) (Total for Question 8 is 4 marks)
Show worked solution
  1. Convert standard form
    \[ 520\text{ million}=520000000=5.2\times 10^8 \]
    \[ 8.79\times 10^{-5}=0.0000879 \]
  2. Convert standard form

    For part (c),

    \[ (5\times 10^{42})(7\times 10^{-180}) =35\times 10^{-138} \]
  3. Write this in standard form

    Write this in standard form:

    \[ 35\times 10^{-138}=3.5\times 10^{-137} \]
Final answer

(a) \(5.2\times 10^8\), (b) \(0.0000879\), (c) \(3.5\times 10^{-137}\)

26. Jun 2024 P2HR · Q9

Medium4 marks
Jun 2024 P2HR Question 9 (4 marks): Powers, Roots and Standard Form – 9 (a) Write 6.04 × 105 as an ordinary number.  (1) (b) Write 0.000 07 in standard form.  (1) (c) Work out 7 6 10 4 10 2 10 10 5 4 . Give your answer in standard form.  (2) (Total for Question 9 is 4 marks)
Show worked solution
  1. Convert standard form
    \[ 6.04\times 10^5=604000 \]
    \[ 0.00007=7\times 10^{-5} \]
  2. Convert standard form

    For part (c), first simplify the denominator:

    \[ 4\times 10^5-2\times 10^4=400000-20000=380000 \]
    \[ 380000=3.8\times 10^5 \]
  3. Convert standard form

    Then

    \[ \frac{7.6\times 10^{10}}{3.8\times 10^5} =2\times 10^5 \]
Final answer

(a) \(604000\), (b) \(7\times 10^{-5}\), (c) \(2\times 10^5\)

27. Nov 2023 P1H · Q9

Medium4 marks
Nov 2023 P1H Question 9 (4 marks): Powers, Roots and Standard Form – 9 (a) Write 5 87 10 4 . as an ordinary number. (1) (b) Write 84 000 000 in standard form. (1) The number of neurons in a human brain is 8 5 1010 . × The number of neurons in a monkey brain is 1 47 109 . × The number
Show worked solution
  1. Convert standard form
    \[ 5.87\times 10^{-4}=0.000587 \]
    \[ 84000000=8.4\times 10^7 \]
  2. Convert standard form

    For part (c),

    \[ K=\frac{8.5\times 10^{10}}{1.47\times 10^9} \]
    \[ K=\frac{8.5}{1.47}\times 10 \]
    \[ K=57.823\ldots \]
  3. Convert standard form

    Correct to one decimal place,

    \[ K=57.8 \]
Final answer

(a) \(0.000587\), (b) \(8.4\times 10^7\), (c) \(57.8\)

28. Jun 2022 P2HR · Q9

Medium4 marks
Jun 2022 P2HR Question 9 (4 marks): Powers, Roots and Standard Form – 9 The table gives information about the population, correct to 2 significant figures, of each of five cities in 2018 City Population (2018) Ahmedabad 7.7 × 10 6 Barcelona 5.5 × 10 6 Chicago 8.8 × 10 6 Lagos 1.3 × 10 7 To
Show worked solution
  1. Convert standard form
    \[ 8.8\times 10^6=8800000 \]
  2. The least population in the

    The least population in the table is Barcelona, with

    \[ 5.5\times 10^6 \]
  3. Convert standard form

    For the difference between Tokyo and Ahmedabad:

    \[ 3.7\times 10^7-7.7\times 10^6 \]
    \[ 37000000-7700000=29300000 \]
    \[ 29300000=2.93\times 10^7 \]
  4. Correct to 2 significant figures

    Correct to 2 significant figures:

    \[ 2.9\times 10^7 \]
Final answer

(a) \(8800000\), (b) Barcelona, (c) \(2.9\times 10^7\)

29. Jan 2022 P1HR · Q8

Medium4 marks
Jan 2022 P1HR Question 8 (4 marks): Powers, Roots and Standard Form – 8 (a) Write 5 × 104 as an ordinary number. (1) (b) Write 0.000 06 in standard form. (1) (c) Work out (4 × 10512 ) ÷ (1.6 × 10700 ) Give your answer in standard form. (2) (Total for Question 8 is 4 marks)
Show worked solution
  1. Convert standard form
    \[ 5\times 10^4=50000 \]
    \[ 0.00006=6\times 10^{-5} \]
  2. Convert standard form

    For part (c),

    \[ \frac{4\times 10^{512}}{1.6\times 10^{700}} =\frac{4}{1.6}\times 10^{512-700} \]
    \[ =2.5\times 10^{-188} \]
Final answer

(a) \(50000\), (b) \(6\times 10^{-5}\), (c) \(2.5\times 10^{-188}\)

30. May 2021 P1H · Q8

Medium4 marks
May 2021 P1H Question 8 (4 marks): Powers, Roots and Standard Form – 8 The table shows the populations of five countries. Country Population China 1.4 × 109 Germany 8.2 × 107 Sweden 9.9 × 106 Fiji 9.1 × 105 Malta 4.3 × 105 (a) Work out the difference between the population of China and th
Show worked solution
  1. Convert standard form

    For part (a),

    \[ 1.4\times 10^9=1400000000 \]
    \[ 8.2\times 10^7=82000000 \]
  2. Difference

    Difference:

    \[ 1400000000-82000000=1318000000 \]
    \[ 1318000000=1.318\times 10^9 \]
  3. Convert standard form

    For part (b),

    \[ \text{population of Fiji}=\frac{1}{k}\times \text{population of Sweden} \]
  4. Convert standard form

    So

    \[ k=\frac{\text{population of Sweden}}{\text{population of Fiji}} \]
    \[ k=\frac{9.9\times 10^6}{9.1\times 10^5}=10.879\ldots \]
  5. Convert standard form

    To the nearest whole number,

    \[ k=11 \]
Final answer

(a) \(1.318\times 10^9\), (b) \(11\)

31. Jan 2021 P2H · Q6

Medium4 marks
Jan 2021 P2H Question 6 (4 marks): Powers, Roots and Standard Form – 6 (a) Write down the value of y0 (1) (b) Work out 9 6 10 6 4 10 3 2 10 141 140 16 . . . × + × × Give your answer in standard form. (3) (Total for Question 6 is 4 marks)
Show worked solution
  1. Convert standard form

    Any non-zero value to the power \(0\) is \(1\), so

    \[ y^0=1 \]
  2. Convert standard form

    For part (b),

    \[ 9.6\times 10^{141}+6.4\times 10^{140} =9.6\times 10^{141}+0.64\times 10^{141} \]
    \[ =10.24\times 10^{141}=1.024\times 10^{142} \]
  3. Convert standard form

    Now divide:

    \[ \frac{1.024\times 10^{142}}{3.2\times 10^{16}} =0.32\times 10^{126} \]
    \[ =3.2\times 10^{125} \]
Final answer

(a) \(1\), (b) \(3.2\times 10^{125}\)

32. May-Nov 2020 P1H · Q5

Medium4 marks
May-Nov 2020 P1H Question 5 (4 marks): Powers, Roots and Standard Form – 5 (a) Write 5.7 × 10 -3 as an ordinary number. (1) (b) Write 800 000 in standard form. (1) (c) Work out 3 10 2 7 10 6 10 5 4 2 × - × × - . (2) (Total for Question 5 is 4 marks)
Show worked solution
  1. Convert standard form
    \[ 5.7\times 10^{-3}=0.0057 \]
    \[ 800000=8\times 10^5 \]
  2. Convert standard form

    For part (c), first simplify the numerator:

    \[ 3\times 10^5-2.7\times 10^4=300000-27000=273000 \]
  3. Convert standard form

    Then

    \[ \frac{273000}{6\times 10^{-2}}=\frac{273000}{0.06}=4550000 \]
    \[ 4550000=4.55\times 10^6 \]
Final answer

(a) \(0.0057\), (b) \(8\times 10^5\), (c) \(4.55\times 10^6\)

33. May 2026 4WM1H · Q3

Hard5 marks
May 2026 4WM1H Question 3 (5 marks): Powers, Roots and Standard Form – 3 20 20 70 8 20 n (a) Find the value of n n = (1) 36 36 80 5 36 m (b) Find the value of m m = (1) 52 52 4 16 p (c) Find the value of p p = (1) In a race, Dani ran a distance of 100 metres, correct to the nearest me
Show worked solution
  1. (a) Multiply: add powers
    \[20^{70}\times20^{8}=20^{78}\Rightarrow n=78\]
  2. (b) Divide: subtract powers
    \[36^{80}\div36^{5}=36^{75}\Rightarrow m=75\]
  3. (c) Power of a power: multiply
    \[(52^4)^{16}=52^{64}\Rightarrow p=64\]
  4. (d) Lower bound

    Nearest metre means \(\pm0.5\) m: lower bound \(=99.5\) m.

  5. (e) Upper bound

    One decimal place means \(\pm0.05\) s: upper bound \(=16.35\) s.

Final answer

(a) \(78\) (b) \(75\) (c) \(64\) (d) \(99.5\) m (e) \(16.35\) s

34. Nov 2021 P2H · Q9

Hard5 marks
Nov 2021 P2H Question 9 (5 marks): Powers, Roots and Standard Form – 9 A rainwater tank contains 2.4 ́ 107 raindrops. The rainwater tank also contains 1.75 ́ 106 bacteria. (a) Work out the number of bacteria per raindrop in the tank. Give your answer in standard form correct to 2 signific
Show worked solution
  1. Convert standard form

    For part (a), divide the number of bacteria by the number of raindrops:

    \[ \frac{1.75\times 10^6}{2.4\times 10^7} =\frac{1.75}{2.4}\times 10^{-1} \]
    \[ =0.072916\ldots \]
  2. Correct to 2 significant figures

    Correct to 2 significant figures:

    \[ 7.3\times 10^{-2} \]
  3. Convert standard form

    For part (b), one drop contains \(5.01\times 10^{21}\) atoms.

  4. Convert standard form

    The number of molecules in one drop is

    \[ \frac{5.01\times 10^{21}}{3}=1.67\times 10^{21} \]
  5. Convert standard form

    There are \(2.4\times 10^7\) drops, so the number of molecules in the tank is

    \[ (1.67\times 10^{21})(2.4\times 10^7) =4.008\times 10^{28} \]
  6. Correct to one significant figure

    Correct to one significant figure:

    \[ 4\times 10^{28} \]
Final answer

(a) \(7.3\times 10^{-2}\), (b) \(4\times 10^{28}\) molecules

35. May 2023 P1H · Q15

Hard6 marks
May 2023 P1H Question 15 (6 marks): Powers, Roots and Standard Form – 15 (a) 2 8 16 2 3 3 2 n Work out the value of n Show your working clearly. n = (3) (b) Find 4% of 4.5 × 10 157 Give your answer in standard form. (3) (Total for Question 15 is 6 marks)
Show worked solution
  1. Convert standard form
    \[ \sqrt{2}=2^{\frac12} \]
    \[ 8^3=(2^3)^3=2^9 \]
    \[ 16^{\frac32}=(2^4)^{\frac32}=2^6 \]
  2. Convert standard form

    So

    \[ \frac{8^3}{16^{\frac32}}=\frac{2^9}{2^6}=2^3 \]
  3. Convert standard form

    Therefore

    \[ \sqrt{2}\div \frac{8^3}{16^{\frac32}} =2^{\frac12}\div 2^3 \]
    \[ =2^{\frac12-3}=2^{-\frac52} \]
  4. Convert standard form

    So

    \[ n=-\frac52 \]
  5. Convert standard form

    For part (b),

    \[ 4\%=0.04=4\times 10^{-2} \]
    \[ 0.04(4.5\times 10^{157})=18\times 10^{155} \]
    \[ =1.8\times 10^{156} \]
Final answer

(a) \(n=-\frac52\), (b) \(1.8\times 10^{156}\)

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Dr Eslam Ahmed
Dr Eslam Ahmed

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