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EXERCISE \( 4.2 \) 1. Find the cubes of: (i) \( -11 \) (ii) \( -12 \) (iii) \( -21 \) 2. Which of the following numbers are cubes of negative integers (i) \( -64 \) (ii) \( -1056 \) (iii) \( -2197 \) (iv) \( -2744 \) (v) \( -42875 \) 3. Show that the following integers are cubes of negative integers. Also, find the integer whose cube is the given integer. (i) \( -5832 \) (ii) \( -2744000 \) 4. Find the cube of: (i) \( \frac{7}{9} \) (ii) \( -\frac{8}{11} \) (iii) \( \frac{12}{7} \) (iv) \( -\frac{13}{8} \) (v) \( 2 \frac{2}{5} \) (vi) \( 3 \frac{1}{4} \) (vii) \( 0.3 \) (viii) \( 1.5 \) (ix) \( 0.08 \) (x) \( 2.1 \) 5. Find which of the following numbers are cubes of rational numbers: (i) \( \frac{27}{64} \) (ii) \( \frac{125}{128} \) (iii) \( 0.001331 \) (iv) \( 0.04 \)

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Used formula (ab)n = anbn

1. (i) (-11)3 = (-1)3 x 113 = -121 x 11 = -1331

(ii) (-12)3 = -123 = -144 x 12 = -1728

(iii) (-21)3 = -213 = -441 x 21 = -9261

2. (i) -64 = -43 = (-4)3

(ii) -1056 = -25 x 33

i.e. -1056 is not a cube of negative integer.

(iii) -2197 = -133

(iv) -2744 = -143

(v) -42875 = -353

3. (i) -5832 = -23 x 36 = (-2 x 32)3 = (-18)3

(ii) -2744000 = -2744 x 103 = (-2 x 7 x 10)3 = (-140)3

4. (i) (7/9)3 \(=\frac{7^3}{9^3}\) = 343/729

(ii) (-8/11)3 \(=\frac{-8^3}{11^3}\) = -512/1331

(iii) (12/7)3 \(=\frac{12^3}{7^3}\) = 1728/343

(iv) (-13/8)3 \(=\frac{-13^3}{8^3}\) = -2197/512

(v) (2 2/5)3 = (12/5)3 \(=\frac{12^3}{5^3}\) = 1728/125 = 13 103/125

(vi) (3 1/4)3 = (13/4)\(=\frac{13^3}{4^3}\) = 2197/64 = 34 21/64

(vii) (0.3)3 = (3 x 10-1)3 = 33 x 10-3 = 27 x 10-3 = 0.027

(viii) (1.5)3 = (15 x 10-1)3 = 153 x 10-3 = 3375 x 10-3 = 3.375

(ix) (0.08)3 = (8 x 10-2)3 = 83 x 10-6 = 512 x 10-6 = 0.000512

(x) (2.1)3 = (21 x 10-1)3 = 213 x 10-3 = 9261 x 10-3 = 9.261

(5) (i) 27/64 \(=\frac{3^3}{4^3}\) = (3/4)3

(ii) 125/128 \(=\frac{125}{64\times2}=\frac{5^3}{4^3\times2}\) = 1/2 (5/4)3

Not a cube of rational number.

(iii) 0.001331 = 1331 x 10-6 = 113 x (10-2)3 = (11 x 10-2)3

= (0.11)3

(iv) 0.04 = 4/100 = 1/25 = (1/5)2

Not a cube of rational number.

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