Express the following decimals in the form p/q.

(i)0.32 (ii) 0.123 (iii) 0.00352

(i)0.32 (ii) 0.123 (iii) 0.00352

Let x = 0.32

Clearly, there is just one digit on the right side of the decimal point which is without bar. So, we multiply both sides of x by 10 so that only the repeating decimals is ledt on the right side of the decimal point.

âˆ´10x = 3.2

=>10x = 3 + 0.2

10x = 3 + 2/9 [0.2 = 2/9]

10x = (9 x 3+2)/9 => 10x = 29/9 => x= 29/90

(ii)Let x = 0.123

Clearly, there are two digits on the right side of the decimal point which are without bar.So, we multiply both sides of x by 102 = 100 so that only the repeating decimals is left on the right side of the decimal point.

âˆ´ 100x = 12.3

=> 100x = 12 + 0.3

=> 100x = 12 + 3/9

=> 100x = (12 x 9 +3)/9

=> 100x = (108 + 3)/9 => 100x = 111/9 => x = 111/900

(iii)Let x = 0.00352

Clearly, there are two digits on the right side of the decimal point which are without bar.So, we multiply both sides of x by 103 = 1000 so that only the repeating decimals is left on the right side of the decimal point.

âˆ´ 1000x = 3.52

=> 1000x = 3 + 0.52

=> 1000x = 3 + 52/99

=> 100x = (3 x 99 +52)/99

=> 1000x = (297 +52)/99 => 1000x = 349/99 => x = 349/99000

Clearly, there is just one digit on the right side of the decimal point which is without bar. So, we multiply both sides of x by 10 so that only the repeating decimals is ledt on the right side of the decimal point.

âˆ´10x = 3.2

=>10x = 3 + 0.2

10x = 3 + 2/9 [0.2 = 2/9]

10x = (9 x 3+2)/9 => 10x = 29/9 => x= 29/90

(ii)Let x = 0.123

Clearly, there are two digits on the right side of the decimal point which are without bar.So, we multiply both sides of x by 102 = 100 so that only the repeating decimals is left on the right side of the decimal point.

âˆ´ 100x = 12.3

=> 100x = 12 + 0.3

=> 100x = 12 + 3/9

=> 100x = (12 x 9 +3)/9

=> 100x = (108 + 3)/9 => 100x = 111/9 => x = 111/900

(iii)Let x = 0.00352

Clearly, there are two digits on the right side of the decimal point which are without bar.So, we multiply both sides of x by 103 = 1000 so that only the repeating decimals is left on the right side of the decimal point.

âˆ´ 1000x = 3.52

=> 1000x = 3 + 0.52

=> 1000x = 3 + 52/99

=> 100x = (3 x 99 +52)/99

=> 1000x = (297 +52)/99 => 1000x = 349/99 => x = 349/99000

$\frac{{x}^{\mathrm{a(b-c)}}}{{x}^{\mathrm{b(a-c)}}}\xf7{\left(\frac{{x}^{\mathrm{b}}}{{x}^{\mathrm{a}}}\right)}^{\mathrm{c}}\mathrm{=\; 1}(ii)\frac{{\left({x}^{\mathrm{a+b}}\right)}^{2}{\left({x}^{\mathrm{b+c}}\right)}^{2}{\left({x}^{\mathrm{c+a}}\right)}^{2}}{\mathrm{(xaxbxc)4}}=\; 1$

$\sqrt{{x}^{-1}\mathrm{y}}.\sqrt{{x}^{-1}\mathrm{z}}.\sqrt{{z}^{-1}\mathrm{x}}$

(i) a

(iv) (a/b)

(i) (2/11)

(iii) 2

(i) 5

(i) (625)

(i) a rational number (ii) an irrational number.

(i) $\sqrt{{x}^{-2}{y}^{3}}$ (ii) ${\left({x}^{\mathrm{-2/3}}{y}^{\mathrm{-1/2}}\right)}^{2}$ (iii) ${\left(\sqrt{{x}^{-3}}\right)}^{5}$ (iv) ${\left(\sqrt{\mathrm{x}}\right)}^{\mathrm{-2/3}}\sqrt{{y}^{4}}\xf7\sqrt{{y}^{4}}$

(v) $3\sqrt{{\mathrm{xy}}^{2}\xf7{x}^{2}\mathrm{y}}$ (vi) $4\sqrt{3\sqrt{{x}^{2}}}$

(i)Â 0.4Â Â (ii)Â 0.2 Â Â (iii)Â 0.3

(iv)0.4 Â Â (v)Â 0.5Â Â (vi)0.6

a = 0.1111... = 0.1 and b = 0.1101

(i) (64)

(i) (√2 + 2)

(i)Â 0.15 Â Â Â (ii)Â 0.675Â (iii)Â 0.00026

(i) (√4)

(iv) (25)

Show that 1.272727 = 1.27 can be expressed in the form p/q , where p and q are integers and q ≠0.

(i) 15.75 0 (ii) 8.0025Â (iii) -24.6875