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## Tamilnadu Samacheer Kalvi 9th Maths Solutions Chapter 3 Algebra Ex 3.9

Question 1.

Find the GCD for the following:

(i) p^{5}, p^{11}, p^{9}

(ii) 4x^{3}, y^{3}, z^{3}

(iii) 9 a^{2} b^{2} c^{3}, 15 a^{3} b^{2} c^{4}

(iv) 64x^{8}, 240x^{6}

(v) ab^{2} c^{3}, a^{2} b^{3} c, a^{3} bc^{2}

(vi) 35x^{5} y^{3} z^{4}, 49x^{2} yz^{3}, 14xy^{2} z^{2}

(vii) 25 ab^{3} c, 100 a^{2} bc, 125 ab

(viii) 3abc, 5xyz, 7pqr

Solution:

(i) p^{5}, p^{11}, p^{9}

p^{5} = p^{5}

p^{9} = p^{5} × p^{4}

p^{11} = p^{5} × p^{6}

G.C.D. of p^{5}, p^{11}, p^{9} = p^{5}

(ii) 4x^{3}, y^{3}, z^{3}

G.C.D. of 4x^{3} = 1 × 4x^{3}

y^{3} = 1 × y^{3}

z^{3} = 1 × z^{3}

∴ G.C.D. = 1

(iii) 9 a^{2} b^{2} c^{3}, 15 a^{3} b^{2} c^{4}

9 a^{2} b^{2} c^{3} = 3 × 3a^{2} b^{2} c^{3}

15 a^{3} b^{2} c^{4} = 3 × 5a^{3} b^{2} c^{4}

G.C.D. = 9 a^{2} b^{2} c^{3}, 15 a^{3} b^{2} c^{4} = 3 × a^{2} b^{2} c^{3} = 3 a^{2} b^{2} c^{3}

(iv) 64x^{8}, 240x^{6}

64x^{8} = 2 × 2 × 2 × 2 × 2 × 2 × x^{6} × x^{8}

240x^{6} = 2 × 2 × 2 × 2 × 3 × 5 x^{6}

G.C.D. of 64x^{8}, 240x^{6} = 2 × 2 × 2 × 2 x^{6} = 16x^{6}

(v) ab^{2} c^{3}, a^{2} b^{3} c, a^{3} bc^{2}

ab^{2} c^{3} = a × b × b × c × c × c

a^{2} b^{3} c = a × a × b × b × b × c

a^{3} bc^{2} = a × a × a × b × c × c

G.C.D. of ab^{2} c^{3}, a^{2} b^{3} c, a^{3} bc^{2} = a × b × c = abc

(vi) 35x^{5} y^{3} z^{4}, 49x^{2} yz^{3}, 14xy^{2} z^{2}

35x^{5} y^{3} z^{4} = 5 × 7 x^{5} y^{3} z^{4}

49x^{2} yz^{3} = 7 × 7 x^{2} y z^{3}

14xy^{2} z^{2} = 2 × 7 x y^{2} z^{2}

G.C.D. is = 7 x y z^{2}

(vii) 25 ab^{3} c, 100 a^{2} bc, 125 ab

25 ab^{3} c = 5 × 5 ab^{3}c

100 a^{2} bc = 2 × 5 × 2 × 5 a^{2} b c

125 ab = 5 × 5 × 5 ab

G.C.D is = 5 × 5 ab = 25ab

(viii) 3abc, 5xyz, 7pqr

3abc = 1 × 3 a bc

5xyz = 1 × 5 x y z

7pqr = 1 × 7 p q r

G.C.D is 1

Question 2.

Find the GCD of the following

(i) (2x + 5), (5x + 2)

(ii) a^{m + 1}, a^{m + 2}, a^{m + 3}

(iii) 2a^{2} + a, 4a^{2} – 1

(iv) 3a^{2}, 5b^{3}, 7c^{4}

(v) x^{4} – 1, x^{2} – 1

(vi) a^{3} – 9ax^{2}, (a – 3x)^{2}

Solution:

(i) (2x + 5), (5x + 2)

(2x + 5) = 1 × (2x + 5)

(5x + 2) = 1 × (5x + 2)

∴ G.C.D = 1

(ii) a^{m + 1}, a^{m + 2}, a^{m + 3}

a^{m + 1} = a^{m} × a^{1}

a^{m + 2} = a^{m} × a^{1} × a^{1}

a^{m + 3} = a^{m} × a^{1} × a^{1} × a^{1}

G.C.G = a^{m} × a^{1} = a^{m + 1}

(iii) 2a^{2} + a, 4a^{2} – 1

2a^{2} + a = a (2a + 1)

4a^{2} – 1 = (2a)^{2} – 1^{2} = (2a + 1) (2a -1)

G.C.D = (2a + 1)

(iv) 3a^{2}, 5b^{3}, 7c^{4}

3a^{2} = 1 × 3a × a

5b^{3} = 1 × 5 b × b × b

7c^{4} = 1 × 7 × c × c × c × c

∴ G.C.D = 1

(v) x^{4} – 1, x^{2} – 1

x^{4} – 1 = (x^{2})^{2} – 1^{2} = (x^{2} + 1) (x^{2} – 1)

= (x^{2} + 1) (x^{2} – 1^{2}) = (x^{2} + 1) (x + 1 ) (x – 1)

x^{2} – 1 = x^{2} – 1^{2} = (x + 1) (x – 1)

G.C.D = (x + 1) (x – 1) = x^{2} – 1

(vi) a^{3} – 9ax^{2}, (a – 3x)^{2}

a^{3} – 9ax^{2} = a (a^{2} – (3x)^{2}) = a (a + 3x) (a – 3x)

(a – 3x)^{2} = (a – 3x) (a – 3x)

G.C.D = (a – 3x)