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Solutions — Full Answer Key

Ecolint Campus des NationsMathematics
Year 9 · 9.7 Indices and Scientific Notation

Solutions · Full Answer Key

Pack A answers · Pack B answers · Problem-solving worked solutions

Pack A — Answers

Bronze
1.32
2.373^7
3.757^5
4.3.2×1043.2 \times 10^4
5.4×10−34 \times 10^{-3}
6.11 and 1/491/49
7.2122^{12}
8.x8x^8
9.5×1045 \times 10^4
10.1 and 5
Silver
11.6×1096 \times 10^9
12.4×1064 \times 10^6
13.x5x^5
14.5
15.3.5×1063.5 \times 10^6
16.5.3×1045.3 \times 10^4
17.1/4=0.251/4 = 0.25
18.8x38x^3
19.9.5×10159.5 \times 10^{15} m
20.4×1064 \times 10^6
Gold
21.x7x^7
22.3.5×1063.5 \times 10^6
23.5
24.6.66×1076.66 \times 10^7
25.1.2×1031.2 \times 10^3
26.0.0036
27.4
28.3×1033 \times 10^3
29.36x836 x^8
30.2×10−4<3×10−3<4×10−32 \times 10^{-4} < 3 \times 10^{-3} < 4 \times 10^{-3}
Platinum
31.x=3x = 3
32.1/41/4
33.3.97×10133.97 \times 10^{13} km
34.x=64x = 64
35.(a) 5×1045 \times 10^4 (b) 8×1058 \times 10^5 (c) 12 hours
36.1/91/9
37.x=5x = 5
38.x=3x = 3 or x=−3x = -3
39.3a33 a^3
40.1010 and 102=10010^2 = 100

Pack B — Answers

Bronze
1.81
2.575^7
3.262^6
4.7.5×1067.5 \times 10^6
5.2.5×10−42.5 \times 10^{-4}
6.11 and 1/251/25
7.5125^{12}
8.x9x^9
9.2.5×10−22.5 \times 10^{-2}
10.1 and 7
Silver
11.2×10102 \times 10^{10}
12.3×1033 \times 10^3
13.x7x^7
14.9
15.6×1036 \times 10^3
16.2.4×1052.4 \times 10^5
17.1/10=0.11/10 = 0.1
18.9x29x^2
19.Same.
20.2.5×1092.5 \times 10^9
Gold
21.x7x^7
22.6×1036 \times 10^3
23.10
24.6.82×1076.82 \times 10^7
25.104=1×10410^4 = 1 \times 10^4
26.405 000
27.9
28.2×1042 \times 10^4
29.Same.
30.3×104<5×104<2×1053 \times 10^4 < 5 \times 10^4 < 2 \times 10^5
Platinum
31.x=±2x = \pm 2
32.1/101/10
33.8.14×10138.14 \times 10^{13} km
34.x=125x = 125
35.Same.
36.1/81/8
37.x=−3x = -3
38.x=2x = 2 or x=−2x = -2
39.8b28 b^2
40.Same.

Problem-solving — Worked Solutions

1Problem 1
Answer
(a) 353^5 (b) x5x^5 (c) 2122^{12} (d) x2x^2
Full working
(a) 353^5. (b) Subtract: x5x^5. (c) Multiply: 2122^{12}. (d) Add: x2x^2.
2Problem 2
Answer
(a) 1.006 (b) 6.66×1076.66 \times 10^7
Full working
(a) 1 + 0.006 = 1.006.

(b) 6.50×107×1.0064≈6.50×107×1.02418≈6.657×107≈6.66×1076.50 \times 10^7 \times 1.006^4 \approx 6.50 \times 10^7 \times 1.02418 \approx 6.657 \times 10^7 \approx 6.66 \times 10^7.
3Problem 3
Answer
(a) 1.2×1051.2 \times 10^5 (b) 4×1054 \times 10^5 (c) 8×10128 \times 10^{12}
Full working
(a) 12×104=1.2×10512 \times 10^4 = 1.2 \times 10^5.

(b) 4×1054 \times 10^5.

(c) 8×10128 \times 10^{12}.
4Problem 4
Answer
(a) 5 (b) 2 (c) 8 (d) 1/91/9
Full working
(a) 25=5\sqrt{25} = 5.

(b) 83=2\sqrt[3]{8} = 2.

(c) (161/4)3=23=8(16^{1/4})^3 = 2^3 = 8.

(d) 1/272/3=1/91/27^{2/3} = 1/9.
5Problem 5
Answer
(a) 5×1045 \times 10^4 (b) 8×1058 \times 10^5 (c) 12
Full working
(a) 5×1045 \times 10^4. (b) 2.5×104×32=8×1052.5 \times 10^4 \times 32 = 8 \times 10^5. (c) 2n>4000⇒n≥122^n > 4000 \Rightarrow n \geq 12.
6Problem 6
Answer
(a) ≈ 1.97×1041.97 \times 10^4 s (b) ≈ 5.5 hours
Full working
(a) Time = distance/speed = 5.9×109/(3×105)≈1.97×1045.9 \times 10^9 / (3 \times 10^5) \approx 1.97 \times 10^4 s.

(b) 1.97×104/3600≈5.471.97 \times 10^4 / 3600 \approx 5.47 hours.
7Problem 7
Answer
(a) x=5x = 5 (b) x=−3x = -3 (c) x=±3x = \pm 3
Full working
(a) 32=2532 = 2^5. (b) 1/125=5−31/125 = 5^{-3}. (c) Even power → both signs.
8Problem 8
Answer
8×103<5×104<9×104<2×1058 \times 10^3 < 5 \times 10^4 < 9 \times 10^4 < 2 \times 10^5
Full working
Compare: 8000, 50000, 90000, 200000.
9Problem 9
Answer
(a) 1.025 (b) ≈ 4.53×1064.53 \times 10^6 (c) Approximately 2029
Full working
(a) 1.025.

(b) 4×106×1.0255≈4×106×1.1314≈4.526×1064 \times 10^6 \times 1.025^5 \approx 4 \times 10^6 \times 1.1314 \approx 4.526 \times 10^6.

(c) Solve 4×106×1.025n>5×106⇒1.025n>1.254 \times 10^6 \times 1.025^n > 5 \times 10^6 \Rightarrow 1.025^n > 1.25. Take logs: n>log⁡1.25/log⁡1.025≈9n > \log 1.25 / \log 1.025 \approx 9. So 2029.
10Problem 10
Answer
(a) ≈ 5.99×10235.99 \times 10^{23} (b) Same
Full working
(a) 1/(1.67×10−24)=(1/1.67)×1024≈0.599×1024=5.99×10231 / (1.67 \times 10^{-24}) = (1/1.67) \times 10^{24} \approx 0.599 \times 10^{24} = 5.99 \times 10^{23}. This is approximately Avogadro's number.
11Problem 11
Answer
(a) 16x1216 x^{12} (b) 3x43 x^4 (c) 3x23 x^2 (d) a2a^2
Full working
(a) 24×x12=16x122^4 \times x^{12} = 16 x^{12}.

(b) Divide coefficients, subtract indices: 3x43 x^4.

(c) Cube root: 3×x23 \times x^2.

(d) a4/2=a2a^{4/2} = a^2.
12Problem 12
Answer
(a) 29=5122^9 = 512 (b) 231≈2.15×1092^{31} \approx 2.15 \times 10^9 (c) 263≈9.22×10182^{63} \approx 9.22 \times 10^{18}
Full working
Each square has 2n−12^{n-1} grains (where nn = square number).

(a) 29=5122^9 = 512.

(b) 231≈2.147×1092^{31} \approx 2.147 \times 10^9.

(c) 263≈9.22×10182^{63} \approx 9.22 \times 10^{18} grains — more than all the rice ever grown!