ГДЗ до підручника «Алгебра» А.Г. Мерзляка. 8 клас

512. 1) √7(√7 - √28) = √49 - √196 = 7 - 14 = -7;

2) (√18 + √72) • √2 = √36 + √144 = 6 + 12 = 18;

3) (4√3 - √75 + 4) • 3√3 = 12√9 - 3√225 + 12√3 = 36 - 45 + 12√3 = 12√3 - 9;

4) (√600 + √6 - √24) • √6 = √3600 + √36 - √144 = 60 + 6 - 12 = 54.

513. 1) (2 - √3)(√3 + 1) = 2√3 + 2 - 3 - √3 = √3 - 1;

2) (√2 + √5)(2√2 - √5) = 4 - √10 + 2√10 - 5 = √10 - 1;

3) (а + √b)(а - √b) = а2 - (√b2) = а2 - b;

4) (√b - √c)(√b + √c) = (√b)2 - (√c)2 = b - с;

5) (4 + √3)(4 - √3) = 16 - (√3)2 = 16 - 3 = 13;

6) (у - √7)(y + √7) = y2 - (√7)2 = y2 - 7;

7) (4√2 - 2√3)(2√3 + 4√2) = (4√2)2 - (2√3)2 = 16 • 2 - 4 • 3 = 32 - 12 = 20;

8) (m + √n)2 = m2 + 2m√n + n;

9) (√a - √b)2 = a - 2√ab + b;

10) (2 - 3√3)2 = 4 - 12√3 + 27 = 31 - 12√3.

514. 1) (√7 + 3)(3√7 - 1) = 3√49 - √7 + 9√7 - 3 = 21 + 8√7 - 3 = 18 + 8√7;

2) (4√2 - √3)(2√2 + 5√3) = 8√4 + 20√6 - 2√6 - 5√9 =16 + 18√6 - 15 = 1 + 18√6;

3) (√p - q)(√p + q) = (√p)2 - q2 = p - q2;

4) (6 - √13)(6 + √13) = 36 - (√13)2 = 36 - 13 = 23;

5) (√5 - х)(√5 + х) = (√5)2 - х2 = 5 - х2;

6) (√19 + √17)(√19 - √17) = (√19)2 - (√17)2 = 19 - 17 = 2;

7) (√6 + √2)2 = 6 + 2√12 + 2 = 8 + 4√3;

8) (3 - 2√15)2 = 9 - 12√15 + 4 • 15 = 69 - 12√15.

515. 1) (2 + √7)2 - 4√7 = 4 + 4√7 + 7 - 4√7 = 11;

2) (√6 - √3)2 + 6√2 = 6 - 2√18 + 3 + 6√2 = 9 - 6√2 + 6√2 = 9 - 6√2 + 6√2 = 9.

516. 1) (3 + √5)2 - 6√5 = 9 + 6√5 + 5 - 6√5 = 14;

2) (√12 - 2√2)2 + 8√6 = 12 - 4√24 + 8 + 8√6 = 20 - 8√6 + 8√6 = 20.

519. 1) а2 - 3 = а2 - (√3)2 = (а - √3)(а + √3);

2) 4b2 - 2 = (2b)2 - (√2)2 = (2b - √2)(2b + √2);

3) 5 - 6с2 = (√5)2 - (√6c)2 = (√5 - √6с)(√5 + √6с);

4) а - 9 = (√а)2 - 32 =(√а - 3)(√а + 3); а ≥ 0;

5) m - n = (√m)2 - (√n)2 = (√m - √n)(√m + √n); m ≥ 0, n ≥ 0;

6) 16х - 25у = (4√х)2 - (5√у)2 = (4√х - 5√y)(4√х + 5√y); х ≥ 0, у ≥ 0;

7) а - 2√а + 1 = (√а - 1)2;

8) 4m - 28√mn + 49n = (2√m)2 - 2 • 2√m • 7√n + (7√n)2 = (2√m - 7√n)2;

9) b + 6√b + 9 = (√b)2 + 2√b • 3 + 32 = (√b + 3)2;

10) 3 + 2√3c + с = (√3)2 + 2 • √3 • √с + (√c)2 = (√3 + √c)2;

11) 2 + √2 = (√2)2 + √2 = √2(√2 + 1);

12) 6√7 - 7 = 6√7 - (√7)2 = √7(6 - √7);

13) а - √а = (√а)2 - √а = √а(√а - 1);

14) √b + √3b = √b(1 + √3);

15) √15 - √5 = √5(√3 - 1).

520. 1) 15 - х2 = (√15)2 - х2 = (√15 - х)(√15 + х);

2) 49х2 - 2 = (7х)2 - (√2)2 = (7х - √2)(7х + √2);

3) 36p - 64q = (6√p)2 - (8√q)2 = (6√р - 8√q)(6√р + 8√q);

4) с - 100 = (√с - 10)(√с + 10), с ≥ 0;

5) а - 8b√a + 16b2 =(√а - 4b)2;

6) m + 2√mn + n = (√m + √n)2, m ≥ 0, n ≥ 0;

7) а - 4√а + 4 = (√а - 2)2;

8) 5 + √5 = (√5)2 + √5 = √5(√5 + 1);

9) √3p - p = √3 • √р - (√р)2 = √р(√3 - √р);

10) √12 + √32 = √4(√3 + √8) = 2(√3 + 2√2).