Chapter 1: Number Systems

Exercise 1.4 & 1.5 - Solutions

Exercise 1.4 & 1.5 Solutions

Exercise 1.4

Question 1: Classify as Rational or Irrational

(i) 2 - √5

  • 2 is rational, √5 is irrational
  • Rational - Irrational = Irrational ✓

(ii) (3 + √23) - √23

  • = 3 + √23 - √23 = 3
  • 3 is Rational ✓

(iii) 2777\frac{2\sqrt{7}}{7\sqrt{7}}

  • = 27\frac{2}{7} (the √7 cancels!)
  • Rational ✓

(iv) 12\frac{1}{\sqrt{2}}

  • = 22\frac{\sqrt{2}}{2} after rationalizing
  • Still has √2, so Irrational ✓

(v) 2π

  • π is irrational, 2 is rational (≠0)
  • Rational × Irrational = Irrational ✓

Question 5: Rationalize Denominators

(i) 17\frac{1}{\sqrt{7}} =17×77=77= \frac{1}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{\sqrt{7}}{7}

(ii) 17−6\frac{1}{\sqrt{7} - \sqrt{6}} =7+6(7)2−(6)2=7+61=7+6= \frac{\sqrt{7} + \sqrt{6}}{(\sqrt{7})^2 - (\sqrt{6})^2} = \frac{\sqrt{7} + \sqrt{6}}{1} = \sqrt{7} + \sqrt{6}

Exercise 1.5: Laws of Exponents

Question 1: Find

(i) 641/264^{1/2} = √64 = 8 ✓

(ii) 321/532^{1/5} = ⁵√32 = ⁵√(2⁵) = 2 ✓

(iii) 1251/3125^{1/3} = ³√125 = ³√(5³) = 5 ✓

Question 2: Find

(i) 93/29^{3/2} Method: am/n=(an)ma^{m/n} = (\sqrt[n]{a})^m = (9)3=33(\sqrt{9})^3 = 3^3 = 27 ✓

(ii) 322/532^{2/5} = (325)2=22(\sqrt[5]{32})^2 = 2^2 = 4 ✓

(iii) 163/416^{3/4} = (164)3=23(\sqrt[4]{16})^3 = 2^3 = 8 ✓

(iv) 125−1/3125^{-1/3} = 11251/3=15\frac{1}{125^{1/3}} = \frac{1}{5} ✓

Question 3: Simplify

(i) 22/3×21/52^{2/3} \times 2^{1/5} Using am×an=am+na^m \times a^n = a^{m+n}: = 22/3+1/5=210/15+3/15=213/152^{2/3 + 1/5} = 2^{10/15 + 3/15} = 2^{13/15} ✓

(iv) 71/2×81/27^{1/2} \times 8^{1/2} Using am×bm=(ab)ma^m \times b^m = (ab)^m: = (7×8)1/2=561/2=56=214(7 \times 8)^{1/2} = 56^{1/2} = \sqrt{56} = 2\sqrt{14} ✓