Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
15 questions · official format, shuffled order
1 (2.5 × 10−3) × (8 × 106) = ?
Show worked solution
2.5 × 8 = 20 and −3 + 6 = 3, so 20 × 103 = 2 × 104. (10−18 multiplies the powers instead of adding them.)
2 Simplify (3x2y)3 ÷ 9x4y2.
Show worked solution
(3x2y)3 = 27x6y3, and 27x6y3 ÷ 9x4y2 = 3x2y. (27x2y forgets to divide the 27 by 9.)
3 0.000 7095 is rounded to 3 significant figures and written in scientific notation. What is the result?
Show worked solution
The next digit is 5, so 709 rounds up to 710: 0.000 710 = 7.10 × 10−4. (7.1 × 10−4 is the same size but shows only 2 significant figures.)
4 3−2 + 3−1 = ?
Show worked solution
3−2 = 1/9 and 3−1 = 3/9, so the sum is 4/9. (3−3 adds the indices, which only works when the powers are multiplied.)
5 If y = 2, what is the value of (3y)0 + 3y0 + 3−1 × 9?
Show worked solution
(3y)0 = 1, 3y0 = 3 × 1 = 3, and 3−1 × 9 = 3. Total 7.
6 Which is the best estimate of 0.0489 × 61 200?
Show worked solution
0.05 × 60 000 = 3000, so 3000. (The exact product is 2992.68.)
7 Which of these is the largest?
Show worked solution
They are 0.0042, 0.0039, 0.0395 and 0.0041. The largest is 3.95 × 10−2 — its power, −2, beats −3.
8 Simplify 12a−3b2 ÷ 4a2b−1, giving the answer with positive indices.
Show worked solution
12 ÷ 4 = 3, a−3 − 2 = a−5 and b2 − (−1) = b3, giving 3a−5b3 = 3b3/a5.
9 If 4x = 83, what is x?
Show worked solution
(22)x = (23)3, so 22x = 29 and 2x = 9: x = 4.5.
10 A hard drive holds 4 × 1012 bytes. One photo uses 2.5 × 106 bytes. How many photos fit on the drive?
Show worked solution
4 ÷ 2.5 = 1.6 and 12 − 6 = 6: 1.6 × 106 photos. (6.25 × 105 divides the wrong way round.)
11 Three statements about 2−3:
1 2−3 = 0.125.
2 (2−3)2 = 2−9.
3 2−3 × 25 = 4.
Which statements are correct?
1 2−3 = 0.125.
2 (2−3)2 = 2−9.
3 2−3 × 25 = 4.
Which statements are correct?
Show worked solution
2−3 = 1/8 = 0.125 (1 ✓). (2−3)2 = 2−6, not 2−9 (2 ✗). 2−3+5 = 22 = 4 (3 ✓). 1 and 3.
12 A length, rounded to 2 significant figures, is 0.0063 m. Which of these could be the true length?
Show worked solution
Only 0.00634 lies from 0.006 25 to below 0.006 35. (0.006249 rounds down to 0.0062.)
13 Which is the best estimate of (6.1 × 104)2?
Show worked solution
62 = 36 and (104)2 = 108, so about 36 × 108 = 3.6 × 109. (1016 raises 4 to the power 2 instead of doubling it; 1.2 × 105 doubles instead of squaring.)
14 Simplify (a4)3 ÷ (a2)5 × a−1.
Show worked solution
a12 ÷ a10 × a−1 = a12−10−1 = a1 = a.
15 The mass of a proton is about 1.67 × 10−27 kg.
1 One million protons have a mass of about 1.67 × 10−21 kg.
2 That is more than 10−20 kg.
3 1027 protons have a mass of about 1.67 kg.
Which statements are correct?
1 One million protons have a mass of about 1.67 × 10−21 kg.
2 That is more than 10−20 kg.
3 1027 protons have a mass of about 1.67 kg.
Which statements are correct?
Show worked solution
106 × 10−27 = 10−21 (1 ✓). 10−21 is smaller than 10−20, so 1.67 × 10−21 is less than 10−20 (2 ✗). 1027 × 10−27 = 100 = 1 (3 ✓). 1 and 3.