3.3. SIMPLIFICATION OF NUMBERS WITH SCIENTIFIC NOTATION
SOLVE
1. Simplify and write the result in scientific notation:
(a) (4 × 10⁵) + (5 × 10⁴)
Solution
(4 × 10⁵) + (5 × 10⁴)
= (4 × 100,000) + (5 × 10,000)
= 400,000 + 50,000
= 450,000
= 4.5 × 10⁵
(b) (9.8 × 10⁵) + (1.5 × 10⁶)
Solution
(9.8 × 10⁵) + (1.5 × 10⁶)
= (9.8 × 100,000) + (1.5 × 1,000,000)
= 980,000 + 1,500,000
= 2,480,000
= 2.48 × 10⁶
(c) (4.5 × 10⁴) + (1.2 × 10⁴)
Solution
(4.5 × 10⁴) + (1.2 × 10⁴)
= (4.5 × 10,000) + (1.2 × 10,000)
= 45,000 + 12,000
= 57,000
= 5.7 × 10⁴
(d) (7.8 × 10³) + (2.1 × 10⁴)
Solution
(7.8 × 10³) + (2.1 × 10⁴)
= (7.8 × 1,000) + (2.1 × 10,000)
= 7,800 + 21,000
= 28,800
= 2.88 × 10⁴
(e) (2.3 × 10⁻³) + (5.6 × 10⁻⁴)
Solution
(2.3 × 10⁻³) + (5.6 × 10⁻⁴)
= 0.0023 + 0.00056
= 0.00286
= 2.86 × 10⁻³
(f) (9 × 10⁻⁶) + (1.2 × 10⁻⁶)
Solution
(9 × 10⁻⁶) + (1.2 × 10⁻⁶)
= 0.000009 + 0.0000012
= 0.0000102
= 1.02 × 10⁻⁵
(g) (1.2 × 10⁻³) + (3.4 × 10⁻⁴)
Solution
(1.2 × 10⁻³) + (3.4 × 10⁻⁴)
= 0.0012 + 0.00034
= 0.00154
= 1.54 × 10⁻³
(h) (1.2 × 10⁻⁴) + (3.4 × 10⁻⁵)
Solution
(1.2 × 10⁻⁴) + (3.4 × 10⁻⁵)
= 0.00012 + 0.000034
= 0.000154
= 1.54 × 10⁻⁴
(i) (2.4 × 10⁻⁶) + (3.7 × 10⁻⁸)
Solution
(2.4 × 10⁻⁶) + (3.7 × 10⁻⁸)
= 0.0000024 + 0.000000037
= 0.000002437
= 2.437 × 10⁻⁶
2. Simplify and write the result in scientific notation:
(a) (8.9 × 10⁷) - (2.1 × 10⁶)
Solution
(8.9 × 10⁷) - (2.1 × 10⁶)
= (8.9 × 10,000,000) - (2.1 × 1,000,000)
= 89,000,000 - 2,100,000
= 86,900,000
= 8.69 × 10⁷
(b) (4.6 × 10⁴) - (8.9 × 10³)
Solution
(4.6 × 10⁴) - (8.9 × 10³)
= (4.6 × 10,000) - (8.9 × 1,000)
= 46,000 - 8,900
= 37,100
= 3.71 × 10⁴
(c) (7 × 10²) - (2.5 × 10¹)
Solution
(7 × 10²) - (2.5 × 10¹)
= (7 × 100) - (2.5 × 10)
= 700 - 25
= 675
= 6.75 × 10²
(d) (6.7 × 10³) - (5.4 × 10²)
Solution
(6.7 × 10³) - (5.4 × 10²)
= (6.7 × 1,000) - (5.4 × 100)
= 6,700 - 540
= 6,160
= 6.16 × 10³
(e) (6.7 × 10⁵) - (1.2 × 10⁴)
Solution
(6.7 × 10⁵) - (1.2 × 10⁴)
= (6.7 × 100,000) - (1.2 × 10,000)
= 670,000 - 12,000
= 658,000
= 6.58 × 10⁵
(f) (2.5 × 10⁻⁶) - (6.4 × 10⁻⁷)
Solution
(2.5 × 10⁻⁶) - (6.4 × 10⁻⁷)
= 0.0000025 - 0.00000064
= 0.00000186
= 1.86 × 10⁻⁶
(g) (9 × 10⁻⁵) - (3 × 10⁻⁶)
Solution
(9 × 10⁻⁵) - (3 × 10⁻⁶)
= 0.00009 - 0.000003
= 0.000087
= 8.7 × 10⁻⁵
(h) (2.3 × 10⁻²) - (5.6 × 10⁻³)
Solution
(2.3 × 10⁻²) - (5.6 × 10⁻³)
= 0.023 - 0.0056
= 0.0174
= 1.74 × 10⁻²
(i) (6.6 × 10⁻⁵) - (9.31 × 10⁻⁹)
Solution
(6.6 × 10⁻⁵) - (9.31 × 10⁻⁹)
= 0.000066 - 0.00000000931
= 0.00006599069
= 6.599069 × 10⁻⁵
3. Simplify and write the result in scientific notation:
(a) (6.2 × 10³) × (2.3 × 10²)
Solution
(6.2 × 10³) × (2.3 × 10²)
= (6.2 × 2.3) × (10³ × 10²)
= 14.26 × 10^{3+2}
= 14.26 × 10⁵
= 1.426 × 10⁶
(b) (8.9 × 10⁻⁵) × (5 × 10⁷)
Solution
(8.9 × 10⁻⁵) × (5 × 10⁷)
= (8.9 × 5) × (10⁻⁵ × 10⁷)
= 44.5 × 10^{-5+7}
= 44.5 × 10²
= 4.45 × 10³
(c) (1.2 × 10³) × (3.4 × 10⁵)
Solution
(1.2 × 10³) × (3.4 × 10⁵)
= (1.2 × 3.4) × (10³ × 10⁵)
= 4.08 × 10^{3+5}
= 4.08 × 10⁸
(d) (5.4 × 10⁴) × (2.1 × 10⁴)
Solution
(5.4 × 10⁴) × (2.1 × 10⁴)
= (5.4 × 2.1) × (10⁴ × 10⁴)
= 11.34 × 10^{4+4}
= 11.34 × 10⁸
= 1.134 × 10⁹
(e) (2.4 × 10⁵) × (6.8 × 10⁻³)
Solution
(2.4 × 10⁵) × (6.8 × 10⁻³)
= (2.4 × 6.8) × (10⁵ × 10⁻³)
= 16.32 × 10^{5-3}
= 16.32 × 10²
= 1.632 × 10³
(f) (8 × 10⁶) × (1.2 × 10⁻²)
Solution
(8 × 10⁶) × (1.2 × 10⁻²)
= (8 × 1.2) × (10⁶ × 10⁻²)
= 9.6 × 10^{6-2}
= 9.6 × 10⁴
(g) (5 × 10⁻⁴) × (2.3 × 10³)
Solution
(5 × 10⁻⁴) × (2.3 × 10³)
= (5 × 2.3) × (10⁻⁴ × 10³)
= 11.5 × 10^{-4+3}
= 11.5 × 10⁻¹
= 1.15 × 10⁰
(h) (5 × 10⁻⁴) × (3.2 × 10⁻³)
Solution
(5 × 10⁻⁴) × (3.2 × 10⁻³)
= (5 × 3.2) × (10⁻⁴ × 10⁻³)
= 16 × 10^{-4-3}
= 16 × 10⁻⁷
= 1.6 × 10⁻⁶
(i) (2.5 × 10⁷) × (1.6 × 10⁻⁴)
Solution
(2.5 × 10⁷) × (1.6 × 10⁻⁴)
= (2.5 × 1.6) × (10⁷ × 10⁻⁴)
= 4 × 10^{7-4}
= 4 × 10³
4. Simplify and write the result in scientific notation:
(a) (1.2 × 10⁴) ÷ (6 × 10²)
Solution
(1.2 × 10⁴) ÷ (6 × 10²)
= \(\frac{(1.2 × 10⁴)}{(6 × 10²)}\)
= \(\frac{1.2}{6} × 10^{4-2}\)
= \(0.2 × 10²\)
= \(2 × 10^{2-1}\)
= \(2 × 10¹\)
(b) (4.5 × 10⁷) ÷ (1.5 × 10³)
Solution
(4.5 × 10⁷) ÷ (1.5 × 10³)
= \(\frac{(4.5 × 10⁷)}{(1.5 × 10³)}\)
= \(\frac{4.5}{1.5} × 10^{7-3}\)
= \(3 × 10⁴\)
(c) (6.7 × 10⁵) ÷ (2 × 10²)
Solution
(6.7 × 10⁵) ÷ (2 × 10²)
= \(\frac{(6.7 × 10⁵)}{(2 × 10²)}\)
= \(\frac{6.7}{2} × 10^{5-2}\)
= \(3.35 × 10³\)
(d) (3 × 10⁶) ÷ (2 × 10³)
Solution
(3 × 10⁶) ÷ (2 × 10³)
= \(\frac{(3 × 10⁶)}{(2 × 10³)}\)
= \(\frac{3}{2} × 10^{6-3}\)
= \(1.5 × 10³\)
(e) (1.2 × 10⁴) ÷ (2.4 × 10²)
Solution
(1.2 × 10⁴) ÷ (2.4 × 10²)
= \(\frac{(1.2 × 10⁴)}{(2.4 × 10²)}\)
= \(\frac{1.2}{2.4} × 10^{4-2}\)
= \(0.5 × 10²\)
= \(5 × 10^{2-1}\)
= \(5 × 10¹\)
(f) (5.6 × 10³) ÷ (2 × 10²)
Solution
(5.6 × 10³) ÷ (2 × 10²)
= \(\frac{(5.6 × 10³)}{(2 × 10²)}\)
= \(\frac{5.6}{2} × 10^{3-2}\)
= \(2.8 × 10¹\)
(g) (3.5 × 10⁻²) ÷ (7 × 10⁻³)
Solution
(3.5 × 10⁻²) ÷ (7 × 10⁻³)
= \(\frac{(3.5 × 10⁻²)}{(7 × 10⁻³)}\)
= \(\frac{3.5}{7} × 10^{-2 - (-3)}\)
= \(0.5 × 10^{-2+3}\)
= \(0.5 × 10¹\)
= \(5 × 10^{1-1}\)
= \(5 × 10⁰\)
(h) (2.7 × 10⁻³) ÷ (3 × 10⁻²)
Solution
(2.7 × 10⁻³) ÷ (3 × 10⁻²)
= \(\frac{(2.7 × 10⁻³)}{(3 × 10⁻²)}\)
= \(\frac{2.7}{3} × 10^{-3 - (-2)}\)
= \(0.9 × 10^{-3+2}\)
= \(0.9 × 10^{-1}\)
= \(9 × 10^{-1 - 1}\)
= \(9 × 10^{-2}\)
(i) (32.08 × 10⁸) ÷ (4.01 × 10⁻³)
Solution
(32.08 × 10⁸) ÷ (4.01 × 10⁻³)
= \(\frac{(32.08 × 10⁸)}{(4.01 × 10⁻³)}\)
= \(\frac{32.08}{4.01} × 10^{8 - (-3)}\)
= \(8 × 10^{8+3}\)
= \(8 × 10^{11}\)
5. Simplify and write the result in scientific notation:
(a) \(\frac{(2.1 × 10³ + 4.9 × 10³)}{(1.5 × 10⁶)}\)
Solution
\(\frac{(2.1 × 10³ + 4.9 × 10³)}{(1.5 × 10⁶)}\)
= \(\frac{10³ (2.1 + 4.9)}{(1.5 × 10⁶)}\)
= \(\frac{10³ × 7}{(1.5 × 10⁶)}\)
= \(\frac{7 × 10³}{1.5 × 10⁶}\)
= \(\frac{7}{1.5} × 10^{3-6}\)
≈ \(4.67 × 10^{-3}\)
(b) \(\frac{(4.4 × 10⁵ + 5.6 × 10⁵)}{(5 × 10⁶)}\)
Solution
\(\frac{(4.4 × 10⁵ + 5.6 × 10⁵)}{(5 × 10⁶)}\)
= \(\frac{10⁵ (4.4 + 5.6)}{(5 × 10⁶)}\)
= \(\frac{10⁵ × 10}{(5 × 10⁶)}\)
= \(\frac{10 × 10⁵}{5 × 10⁶}\)
= \(\frac{10}{5} × 10^{5-6}\)
= \(2 × 10^{-1}\)
= \(0.2\)
= \(2 × 10^{-1}\)
(c) \(\frac{(7.8 × 10⁶ - 1.8 × 10⁶)}{(6 × 10⁵)}\)
Solution
\(\frac{(7.8 × 10⁶ - 1.8 × 10⁶)}{(6 × 10⁵)}\)
= \(\frac{10⁶ (7.8 - 1.8)}{(6 × 10⁵)}\)
= \(\frac{10⁶ × 6}{(6 × 10⁵)}\)
= \(\frac{6 × 10⁶}{6 × 10⁵}\)
= \(\frac{6}{6} × 10^{6-5}\)
= \(1 × 10^{1}\)
= \(10\)
= \(1 × 10^{1}\)
(d) \(\frac{(6.4 × 10⁵ - 1.4 × 10⁵)}{(5 × 10⁴)}\)
Solution
\(\frac{(6.4 × 10⁵ - 1.4 × 10⁵)}{(5 × 10⁴)}\)
= \(\frac{10⁵ (6.4 - 1.4)}{(5 × 10⁴)}\)
= \(\frac{10⁵ × 5}{(5 × 10⁴)}\)
= \(\frac{5 × 10⁵}{5 × 10⁴}\)
= \(\frac{5}{5} × 10^{5-4}\)
= \(1 × 10^{1}\)
= \(10\)
= \(1 × 10^{1}\)
(e) \(\frac{(7.5 × 10⁵ + 4.5 × 10⁵)}{(12 × 10⁴)}\)
Solution
\(\frac{(7.5 × 10⁵ + 4.5 × 10⁵)}{(12 × 10⁴)}\)
= \(\frac{10⁵ (7.5 + 4.5)}{(12 × 10⁴)}\)
= \(\frac{10⁵ × 12}{(12 × 10⁴)}\)
= \(\frac{12 × 10⁵}{12 × 10⁴}\)
= \(\frac{12}{12} × 10^{5-4}\)
= \(1 × 10^{1}\)
= \(10\)
= \(1 × 10^{1}\)
(f) \(\frac{(4.2 × 10⁶) × (4 × 10³)}{(4 × 10⁸)}\)
Solution
\(\frac{(4.2 × 10⁶) × (4 × 10³)}{(4 × 10⁸)}\)
= \(\frac{(4.2 × 4) × (10⁶ × 10³)}{(4 × 10⁸)}\)
= \(\frac{16.8 × 10^{6+3}}{(4 × 10⁸)}\)
= \(\frac{16.8 × 10⁹}{4 × 10⁸}\)
= \(\frac{16.8}{4} × 10^{9-8}\)
= \(4.2 × 10^{1}\)
= \(42\)
= \(4.2 × 10^{1}\)
(g) \(\frac{(2.1 × 10⁶) × (4.0 × 10⁻³)}{(4.2 × 10⁻⁴)}\)
Solution
\(\frac{(2.1 × 10⁶) × (4.0 × 10⁻³)}{(4.2 × 10⁻⁴)}\)
= \(\frac{(2.1 × 4.0) × (10⁶ × 10⁻³)}{(4.2 × 10⁻⁴)}\)
= \(\frac{8.4 × 10^{6-3}}{(4.2 × 10⁻⁴)}\)
= \(\frac{8.4 × 10³}{4.2 × 10⁻⁴}\)
= \(\frac{8.4}{4.2} × 10^{3 - (-4)}\)
= \(2 × 10^{3+4}\)
= \(2 × 10⁷\)
(h) \(\frac{(6.48 × 10⁵)}{(2.4 × 10⁴) × (1.8 × 10⁻²)}\)
Solution
\(\frac{(6.48 × 10⁵)}{(2.4 × 10⁴) × (1.8 × 10⁻²)}\)
= \(\frac{(6.48 × 10⁵)}{(2.4 × 1.8) × (10⁴ × 10⁻²)}\)
= \(\frac{(6.48 × 10⁵)}{(4.32 × 10^{4-2})}\)
= \(\frac{(6.48 × 10⁵)}{(4.32 × 10²)}\)
= \(\frac{6.48}{4.32} × 10^{5-2}\)
= \(1.5 × 10^{3}\)
= \(1500\)
= \(1.5 × 10^{3}\)
(i) \(\frac{(6.2 × 10⁻⁴) × (4 × 10⁵)}{(3.1 × 10⁻³)}\)
Solution
\(\frac{(6.2 × 10⁻⁴) × (4 × 10⁵)}{(3.1 × 10⁻³)}\)
= \(\frac{(6.2 × 4) × (10⁻⁴ × 10⁵)}{(3.1 × 10⁻³)}\)
= \(\frac{24.8 × 10^{-4+5}}{(3.1 × 10⁻³)}\)
= \(\frac{24.8 × 10¹}{(3.1 × 10⁻³)}\)
= \(\frac{24.8}{3.1} × 10^{1 - (-3)}\)
= \(8 × 10^{1+3}\)
= \(8 × 10⁴\)
6. Simplify and write in the usual form:
(a) (3.5 × 10⁴) + (8.7 × 10⁵) - (-2.2 × 10³) + (1.5 × 10³)
Solution:
(3.5 × 10⁴) + (8.7 × 10⁵) - (-2.2 × 10³) + (1.5 × 10³)
= 35,000 + 870,000 + 2,200 + 1,500
= 908,700
(b) (7.5 × 10⁵) - (8.6 × 10⁴) + (2.7 × 10³) + (1.4 × 10⁴)
Solution:
(7.5 × 10⁵) - (8.6 × 10⁴) + (2.7 × 10³) + (1.4 × 10⁴)
= 750,000 - 86,000 + 2,700 + 14,000
= 680,700
7. Simplify and write in usual form:
(a)
\(\frac{(5.5 \times 10^{-4})(6 \times 10^7)}{(3.3 \times 10^{-6})(2 \times 10^4)^2}\)
= \(\frac{33 \times 10^3}{13.2 \times 10^2}\)
= \(2.5 \times 10^1\)
= 25
(b)
\(\frac{(6 \times 10^{-3})(7 \times 10^6)}{(3 \times 10^{-4})(14 \times 10^5)}\)
= \(\frac{42 \times 10^3}{42 \times 10^1}\)
= \(1 \times 10^2\)
= 100
(c)
\(\frac{(15 \times 10^{-3})(2 \times 10^6)}{(5 \times 10^{-4})(15 \times 10^5)}\)
= \(\frac{30 \times 10^3}{75 \times 10^1}\)
= \(0.4 \times 10^2\)
= 40
8. Simplify and express in scientific notation:
(a)
\(\frac{(5 \times 10^{-4})(6 \times 10^7)}{(3 \times 10^{-6})(20 \times 10^{-3})^2}\)
= \(\frac{30 \times 10^3}{1.2 \times 10^{-9}}\)
= \(25 \times 10^{12}\)
= \(2.5 \times 10^{13}\)
(b)
\(\frac{(5 \times 10^6)(4 \times 10^{-7})^2}{(3 \times 10^{-4})(4 \times 10^5)}\)
= \(\frac{80 \times 10^{-8}}{12 \times 10^{-1}}\)
= \(6.\overline{6} \times 10^{-7}\)
= \(6.67 \times 10^{-7}\)
(c)
\(\frac{12,\!000,\!000 \times 0.000003 \times 40,\!000}{0.004 \times 0.003 \times 600,\!000}\)
= \(\frac{(1.2 \times 10^7) \times (3 \times 10^{-6}) \times (4 \times 10^4)}{(4 \times 10^{-3}) \times (3 \times 10^{-3}) \times (6 \times 10^5)}\)
= \(\frac{14.4 \times 10^{5}}{7.2 \times 10^{-1}}\)
= \(2 \times 10^{5-(-1)}\)
= \(2 \times 10^{6}\)
= \(2 \times 10^6\) or 2,000,000
(d)
\(\frac{(6.8 \times 10^5)(3.9 \times 10^{-7})}{7.8 \times 10^{-4}}\)
= \(\frac{26.52 \times 10^{-2}}{7.8 \times 10^{-4}}\)
= \(3.4 \times 10^2\)
(e)
\(\frac{(3.2 \times 10^5)(2.4 \times 10^{-8})}{(6.0 \times 10^{-2})(6.4 \times 10^3)}\)
= \(\frac{7.68 \times 10^{-3}}{3.84 \times 10^2}\)
= \(2 \times 10^{-5}\)
(f)
\(\frac{(3.22 \times 10^3)(4.1 \times 10^{-4})}{(4.07 \times 10^{-5})(0.08 \times 10^4)}\)
= \(\frac{13.202 \times 10^{-1}}{3.256 \times 10^{-2}}\)
= \(4.055 \times 10^1\)
= \(4.06 \times 10^1\)
9. Capacity Problems:
(a) The capacities of two tanks are 4.5 × 10³ liters and 3.5 × 10³ liters. What is the total capacity of both the tanks?
Solution:
Total capacity = (4.5 × 10³) + (3.5 × 10³)
= (4.5 + 3.5) × 10³
= 8.0 × 10³ liters
= 8,000 liters
(b) A water tank can hold 7.5 × 10³ liters of water and another tank can hold 0.5 × 10³ liters of water. What is the sum of the capacities of both water tanks?
Solution:
Total capacity = (7.5 × 10³) + (0.5 × 10³)
= (7.5 + 0.5) × 10³
= 8.0 × 10³ liters
= 8,000 liters
10. Time Calculation Problems:
(a) A rocket has speed 1.6 × 10⁴ km/hour. The distance of the moon from the earth is 3.82 × 10⁶ km. How many hours will it take the rocket to get to the moon?
Solution:
Time= \(\frac{Distance}{Speed}\)
= \(\frac{(3.82 × 10⁶)}{(1.6 × 10⁴)}\)
= \(\frac{(3.82 }{(1.6 × 10⁴)}\)
= (3.82 ÷ 1.6) × 10⁶⁻⁴
= 2.3875 × 10² hours
= 238.75 hours
(b) A rocket travels 5 × 10³ km in one hour. The distance of mars from the earth is 7.8 × 10⁷ km. In how many hours the rocket can reach the mars planet?
Solution:
Time = Distance ÷ Speed
= (7.8 × 10⁷) ÷ (5 × 10³)
= (7.8 ÷ 5) × 10⁷⁻³
= 1.56 × 10⁴ hours
= 15,600 hours
10. Rocket Time Calculation Problems:
(a) A rocket has speed 1.6 × 10⁴ km/hour. The distance of the moon from the earth is 3.82 × 10⁶ km. How many hours will it take the rocket to get to the moon?
Solution:
Time = \(\frac{Distance}{Speed}\)
= \(\frac{3.82 × 10⁶}{1.6 × 10⁴}\) hours
= \(\frac{3.82}{1.6} × 10^{6-4}\) hours
= 2.3875 × 10² hours
= 238.75 hours
(b) A rocket travels 5 × 10³ km/hour. The distance of Mars from the earth is 7.8 × 10⁷ km. In how many hours can the rocket reach Mars?
Solution:
Time = \(\frac{Distance}{Speed}\)
= \(\frac{7.8 × 10⁷}{5 × 10³}\) hours
= \(\frac{7.8}{5} × 10^{7-3}\) hours
= 1.56 × 10⁴ hours
= 15,600 hours
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