Hard GCSE Standard form Questions

Challenging, exam-style GCSE Standard form questions with worked solutions. Stretch yourself on the hardest multiplying standard form, dividing standard form, adding standard form, subtracting standard form problems.

multiplying standard formdividing standard formadding standard formsubtracting standard formword problempowers of standard form
GCSE Foundation34 questionsStep-by-step solutions
Question 1
4 markschallenging
Priya writes 0.50.5 ×\times 10610^{6} as her final answer. Explain why this is not in standard form and give the correct version.
Show worked solution

Worked solution

  1. Recall the rule

    A×10n, 1≤A<10A \times 10^{n},\ 1 \le A < 10

    The front number must be at least 11 and less than 1010.

  2. Look at the front number

    A=0.5A = 0.5

    Here AA is 0.50.5.

  3. Check the range

    0.5<10.5 < 1

    0.50.5 is less than 11, so it breaks the rule.

  4. Conclude it is not standard form

    not standard form\text{not standard form}

    So 0.50.5 x 10610^{6} is not proper standard form.

  5. Rewrite 0.50.5

    0.5=5×10−10.5 = 5 \times 10^{-1}

    Move the point one place right.

  6. Combine the powers

    5×10−1×106=5×1055 \times 10^{-1} \times 10^{6} = 5 \times 10^{5}

    Add the indices: −1+6=5-1 + 6 = 5.

  7. State the correct form

    5×1055 \times 10^{5}

    The proper standard form is 55 x 10510^{5}.

  8. Check AA is in range

    1≤5<101 \le 5 < 10

    Now the front number is in range.

  9. Confirm the value is unchanged

    500000=500000500000 = 500000

    Both forms equal 500000500000.

  10. Answer the question

    5×1055 \times 10^{5}

    So the correct version is 55 x 10510^{5}.

  11. Underline the final answer

    5×1055 \times 10^{5} (since AA must be at least 11).

    Write the answer on its own line so it is clear and easy to mark.

  12. Check it answers the question

    5×1055 \times 10^{5} (since AA must be at least 11).

    Re-reading the question, this is exactly what was required.

  13. State the conclusion in one line

    5×1055 \times 10^{5} (since AA must be at least 11).

    The working above leads directly to this result.

  14. Match to the options

    5×1055 \times 10^{5} (since AA must be at least 11).

    Compare each choice; only the option agreeing with this working is correct.

  15. Sense check

    5×1055 \times 10^{5} (since AA must be at least 11).

    The result is consistent with the checks done along the way.

Answer
5×1055 \times 10^{5} (since AA must be at least 11).
Question 2
4 markschallenging
To work out (33 ×\times 10410^{4}) ×\times (22 ×\times 10510^{5}), what is the correct method and answer?
Show worked solution

Worked solution

  1. Separate the parts

    (3×2)×(104×105)(3 \times 2) \times (10^{4} \times 10^{5})

    Multiply the front numbers together and the powers together.

  2. Multiply the front numbers

    3×2=63 \times 2 = 6

    Multiply the AA parts.

  3. Add the powers

    104×105=10910^{4} \times 10^{5} = 10^{9}

    When multiplying powers of 1010, add the indices, not multiply them.

  4. Reject adding the front and power wrongly

    do not do 4×5\text{do not do } 4 \times 5

    A common error is multiplying the indices; the rule is to add them.

  5. Put it together

    6×1096 \times 10^{9}

    Combine the results.

  6. Check AA is in range

    1≤6<101 \le 6 < 10

    The front number is in range, so no adjusting needed.

  7. Check by expanding

    30000×200000=600000000030000 \times 200000 = 6000000000

    As ordinary numbers this is 66 billion.

  8. Confirm

    6000000000=6×1096000000000 = 6 \times 10^{9}

    Matches the standard-form answer.

  9. State the answer

    6×1096 \times 10^{9}

    So the product is 66 x 10910^{9}.

  10. Conclude

    6×1096 \times 10^{9}

    Multiply the fronts, add the powers.

  11. Underline the final answer

    =6×109= 6 \times 10^{9}

    Write the answer on its own line so it is clear and easy to mark.

  12. Check it answers the question

    6×1096 \times 10^{9}

    Re-reading the question, this is exactly what was required.

  13. State the conclusion in one line

    6×1096 \times 10^{9}

    The working above leads directly to this result.

  14. Match to the options

    6×1096 \times 10^{9}

    Compare each choice; only the option agreeing with this working is correct.

  15. Sense check

    6×1096 \times 10^{9}

    The result is consistent with the checks done along the way.

Answer
6×1096 \times 10^{9}
Question 3
4 markschallenging
Which of these numbers is the largest: 3×1053 \times 10^{5}, 9×1049 \times 10^{4}, 1.2×1061.2 \times 10^{6}, 8×1058 \times 10^{5}?
Show worked solution

Worked solution

  1. Compare the powers of 1010 first

    106>105>10410^{6} > 10^{5} > 10^{4}

    The highest power of 1010 usually points to the largest number.

  2. Spot the highest power

    1.2×1061.2 \times 10^{6}

    Only one number has the power 10610^{6}.

  3. Expand it

    1.2×106=12000001.2 \times 10^{6} = 1200000

    Write it as an ordinary number.

  4. Expand the 10510^{5} numbers

    3×105=300000, 8×105=8000003 \times 10^{5} = 300000,\ 8 \times 10^{5} = 800000

    Both are in the hundred-thousands.

  5. Expand the 10410^{4} number

    9×104=900009 \times 10^{4} = 90000

    This is the smallest so far.

  6. Compare all four

    90000<300000<800000<120000090000 < 300000 < 800000 < 1200000

    List them in order.

  7. Identify the largest

    12000001200000

    1.21.2 x 10610^{6} is the biggest.

  8. State the answer

    1.2×1061.2 \times 10^{6}

    So the largest number is 1.21.2 x 10610^{6}.

  9. Note the lesson

    highest power usually wins\text{highest power usually wins}

    Check the power of 1010 first, then the front number if powers tie.

  10. Conclude

    1.2×1061.2 \times 10^{6}

    The largest is 1.21.2 x 10610^{6}.

  11. Underline the final answer

    =1.2×106= 1.2 \times 10^{6}

    Write the answer on its own line so it is clear and easy to mark.

  12. Check it answers the question

    1.2×1061.2 \times 10^{6}

    Re-reading the question, this is exactly what was required.

  13. State the conclusion in one line

    1.2×1061.2 \times 10^{6}

    The working above leads directly to this result.

  14. Match to the options

    1.2×1061.2 \times 10^{6}

    Compare each choice; only the option agreeing with this working is correct.

  15. Sense check

    1.2×1061.2 \times 10^{6}

    The result is consistent with the checks done along the way.

Answer
1.2×1061.2 \times 10^{6}
Question 4
4 markschallenging
Which is the correct standard form of 0.000560.00056?
Show worked solution

Worked solution

  1. Find the first non-zero digit

    55

    The first significant figure of 0.000560.00056 is 55.

  2. Write AA with one digit before the point

    A=5.6A = 5.6

    Place the decimal point after the 55 to get 5.65.6.

  3. Count how far the point moves

    0.00056→5.60.00056 \rightarrow 5.6

    The point moves 44 places to the right.

  4. Decide the sign of the power

    small number⇒n<0\text{small number} \Rightarrow n < 0

    The number is less than 11, so the power is negative.

  5. Write the power

    n=−4n = -4

    Four places and negative gives 10−410^{-4}.

  6. Write in standard form

    5.6×10−45.6 \times 10^{-4}

    So 0.00056=5.6×10−40.00056 = 5.6 \times 10^{-4}.

  7. Reject 5.65.6 x 10410^{4}

    5.6×104=560005.6 \times 10^{4} = 56000

    A positive power would make it huge, which is wrong.

  8. Reject 5656 x 10−510^{-5}

    56≥1056 \ge 10

    The front number 5656 is not in range.

  9. Check by expanding

    5.6×10−4=0.000565.6 \times 10^{-4} = 0.00056

    Converting back gives the original number.

  10. Conclude

    5.6×10−45.6 \times 10^{-4}

    The correct standard form is 5.65.6 x 10−410^{-4}.

  11. Underline the final answer

    =5.6×10−4= 5.6 \times 10^{-4}

    Write the answer on its own line so it is clear and easy to mark.

  12. Check it answers the question

    5.6×10−45.6 \times 10^{-4}

    Re-reading the question, this is exactly what was required.

  13. State the conclusion in one line

    5.6×10−45.6 \times 10^{-4}

    The working above leads directly to this result.

  14. Match to the options

    5.6×10−45.6 \times 10^{-4}

    Compare each choice; only the option agreeing with this working is correct.

  15. Sense check

    5.6×10−45.6 \times 10^{-4}

    The result is consistent with the checks done along the way.

Answer
5.6×10−45.6 \times 10^{-4}
Question 5
4 markschallenging
Is 4545 ×\times 10310^{3} written in standard form? Explain.
Show worked solution

Worked solution

  1. Recall the rule for standard form

    A×10n, 1≤A<10A \times 10^{n},\ 1 \le A < 10

    The front number AA must be at least 11 and less than 1010.

  2. Look at the front number

    A=45A = 45

    Here the front number is 4545.

  3. Check the range

    45≥1045 \ge 10

    4545 is not less than 1010, so it breaks the rule.

  4. Conclude it is not standard form

    not standard form\text{not standard form}

    So 4545 x 10310^{3} is not in proper standard form.

  5. Fix it: rewrite 4545

    45=4.5×1045 = 4.5 \times 10

    Move the point one place left.

  6. Combine the powers

    4.5×10×103=4.5×1044.5 \times 10 \times 10^{3} = 4.5 \times 10^{4}

    Add the indices: 1+3=41 + 3 = 4.

  7. State the correct form

    4.5×1044.5 \times 10^{4}

    The proper standard form is 4.54.5 x 10410^{4}.

  8. Check AA is in range

    1≤4.5<101 \le 4.5 < 10

    Now the front number is in range.

  9. Confirm the value is unchanged

    45000=4500045000 = 45000

    Both forms equal 4500045000.

  10. Answer the question

    No\text{No}

    No, 4545 x 10310^{3} is not standard form; the correct form is 4.54.5 x 10410^{4}.

  11. Underline the final answer

    No, AA must satisfy 1≤A<101 \le A < 10; the correct form is 4.5×1044.5 \times 10^{4}.

    Write the answer on its own line so it is clear and easy to mark.

  12. Check it answers the question

    No, AA must satisfy 1≤A<101 \le A < 10; the correct form is 4.5×1044.5 \times 10^{4}.

    Re-reading the question, this is exactly what was required.

  13. State the conclusion in one line

    No, AA must satisfy 1≤A<101 \le A < 10; the correct form is 4.5×1044.5 \times 10^{4}.

    The working above leads directly to this result.

  14. Match to the options

    No, AA must satisfy 1≤A<101 \le A < 10; the correct form is 4.5×1044.5 \times 10^{4}.

    Compare each choice; only the option agreeing with this working is correct.

  15. Sense check

    No, AA must satisfy 1≤A<101 \le A < 10; the correct form is 4.5×1044.5 \times 10^{4}.

    The result is consistent with the checks done along the way.

Answer
No, AA must satisfy 1≤A<101 \le A < 10; the correct form is 4.5×1044.5 \times 10^{4}.

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