Страница 31 номер 2.2.23, ГДЗ по алгебре за 7, 8 и 9 класс к задачнику Ткачевой
Разложить на множители по формуле разности квадратов: 1) \(a^2 - 64\); 2) \(b^2 - 36\); 3) \(4m^2 - 25\); 4) \(16n^2 - 81\); 5) \(\frac{9}{16}y^2 - 1\); 6) \(0{,}04a^2 - b^2\); 7) \(a^4 - 0{,}09\); 8) \(b^6 - \frac{16}{25}\); 9) \(-x^6y^8 + \frac{1}{4}\); 10) \(-x^{10}y^6 + \frac{1}{49}\); 11) \(4a^8 - 9b^{12}\); 12) \(36n^4 - 64m^6\).
1) \(a^2 - 64 = a^2 - 8^2 = (a - 8)(a + 8)\);
2) \(b^2 - 36 = b^2 - 6^2 = (b - 6)(b + 6)\);
3) \(4m^2 - 25 = (2m)^2 - 5^2 = (2m - 5)(2m + 5)\);
4) \(16n^2 - 81 = (4n)^2 - 9^2 = (4n - 9)(4n + 9)\);
5) \(\frac{9}{16}y^2 - 1 = \left(\frac{3}{4}y\right)^2 - 1^2 = \left(\frac{3}{4}y - 1\right)\left(\frac{3}{4}y + 1\right)\);
6) \(0{,}04a^2 - b^2 = (0{,}2a)^2 - b^2 = (0{,}2a - b)(0{,}2a + b)\);
7) \(a^4 - 0{,}09 = \left(a^2\right)^2 - 0{,}3^2 = \left(a^2 - 0{,}3\right)\left(a^2 + 0{,}3\right)\);
8) \(b^6 - \frac{16}{25} = \left(b^3\right)^2 - \left(\frac{4}{5}\right)^2 = \left(b^3 - \frac{4}{5}\right)\left(b^3 + \frac{4}{5}\right)\);
9) \(-x^6y^8 + \frac{1}{4} = \frac{1}{4} - x^6y^8 = \left(\frac{1}{2}\right)^2 - \left(x^3y^4\right)^2 = \left(\frac{1}{2} - x^3y^4\right)\left(\frac{1}{2} + x^3y^4\right)\);
10) \(-x^{10}y^6 + \frac{1}{49} = \frac{1}{49} - x^{10}y^6 = \left(\frac{1}{7}\right)^2 - \left(x^5y^3\right)^2 = \left(\frac{1}{7} - x^5y^3\right)\left(\frac{1}{7} + x^5y^3\right)\);
11) \(4a^8 - 9b^{12} = \left(2a^4\right)^2 - \left(3b^6\right)^2 = \left(2a^4 - 3b^6\right)\left(2a^4 + 3b^6\right)\);
12) \(36n^4 - 64m^6 = \left(6n^2\right)^2 - \left(8m^3\right)^2 = \left(6n^2 - 8m^3\right)\left(6n^2 + 8m^3\right)\).
Ответ: 1) \((a - 8)(a + 8)\); 2) \((b - 6)(b + 6)\); 3) \((2m - 5)(2m + 5)\); 4) \((4n - 9)(4n + 9)\); 5) \(\left(\frac{3}{4}y - 1\right)\left(\frac{3}{4}y + 1\right)\); 6) \((0{,}2a - b)(0{,}2a + b)\); 7) \(\left(a^2 - 0{,}3\right)\left(a^2 + 0{,}3\right)\); 8) \(\left(b^3 - \frac{4}{5}\right)\left(b^3 + \frac{4}{5}\right)\); 9) \(\left(\frac{1}{2} - x^3y^4\right)\left(\frac{1}{2} + x^3y^4\right)\); 10) \(\left(\frac{1}{7} - x^5y^3\right)\left(\frac{1}{7} + x^5y^3\right)\); 11) \(\left(2a^4 - 3b^6\right)\left(2a^4 + 3b^6\right)\); 12) \(\left(6n^2 - 8m^3\right)\left(6n^2 + 8m^3\right)\).
Разложить на множители по формуле разности квадратов: 1) \(a^2 - 64\); 2) \(b^2 - 36\); 3) \(4m^2 - 25\); 4) \(16n^2 - 81\); 5) \(\frac{9}{16}y^2 - 1\); 6) \(0{,}04a^2 - b^2\); 7) \(a^4 - 0{,}09\); 8) \(b^6 - \frac{16}{25}\); 9) \(-x^6y^8 + \frac{1}{4}\); 10) \(-x^{10}y^6 + \frac{1}{49}\); 11) \(4a^8 - 9b^{12}\); 12) \(36n^4 - 64m^6\).
Разность квадратов двух выражений равна произведению разности этих выражений и их суммы: \(a^2 - b^2 = (a - b)(a + b)\).
1) \(a^2 - 64 = a^2 - 8^2 = (a - 8)(a + 8)\);
2) \(b^2 - 36 = b^2 - 6^2 = (b - 6)(b + 6)\);
3) \(4m^2 - 25 = (2m)^2 - 5^2 = (2m - 5)(2m + 5)\);
4) \(16n^2 - 81 = (4n)^2 - 9^2 = (4n - 9)(4n + 9)\);
5) \(\frac{9}{16}y^2 - 1 = \left(\frac{3}{4}y\right)^2 - 1^2 = \left(\frac{3}{4}y - 1\right)\left(\frac{3}{4}y + 1\right)\);
6) \(0{,}04a^2 - b^2 = (0{,}2a)^2 - b^2 = (0{,}2a - b)(0{,}2a + b)\);
7) \(a^4 - 0{,}09 = \left(a^2\right)^2 - 0{,}3^2 = \left(a^2 - 0{,}3\right)\left(a^2 + 0{,}3\right)\);
8) \(b^6 - \frac{16}{25} = \left(b^3\right)^2 - \left(\frac{4}{5}\right)^2 = \left(b^3 - \frac{4}{5}\right)\left(b^3 + \frac{4}{5}\right)\);
9) \(-x^6y^8 + \frac{1}{4} = \frac{1}{4} - x^6y^8 = \left(\frac{1}{2}\right)^2 - \left(x^3y^4\right)^2 = \left(\frac{1}{2} - x^3y^4\right)\left(\frac{1}{2} + x^3y^4\right)\);
10) \(-x^{10}y^6 + \frac{1}{49} = \frac{1}{49} - x^{10}y^6 = \left(\frac{1}{7}\right)^2 - \left(x^5y^3\right)^2 = \left(\frac{1}{7} - x^5y^3\right)\left(\frac{1}{7} + x^5y^3\right)\);
11) \(4a^8 - 9b^{12} = \left(2a^4\right)^2 - \left(3b^6\right)^2 = \left(2a^4 - 3b^6\right)\left(2a^4 + 3b^6\right)\);
12) \(36n^4 - 64m^6 = \left(6n^2\right)^2 - \left(8m^3\right)^2 = \left(6n^2 - 8m^3\right)\left(6n^2 + 8m^3\right)\).
Ответ: 1) \((a - 8)(a + 8)\); 2) \((b - 6)(b + 6)\); 3) \((2m - 5)(2m + 5)\); 4) \((4n - 9)(4n + 9)\); 5) \(\left(\frac{3}{4}y - 1\right)\left(\frac{3}{4}y + 1\right)\); 6) \((0{,}2a - b)(0{,}2a + b)\); 7) \(\left(a^2 - 0{,}3\right)\left(a^2 + 0{,}3\right)\); 8) \(\left(b^3 - \frac{4}{5}\right)\left(b^3 + \frac{4}{5}\right)\); 9) \(\left(\frac{1}{2} - x^3y^4\right)\left(\frac{1}{2} + x^3y^4\right)\); 10) \(\left(\frac{1}{7} - x^5y^3\right)\left(\frac{1}{7} + x^5y^3\right)\); 11) \(\left(2a^4 - 3b^6\right)\left(2a^4 + 3b^6\right)\); 12) \(\left(6n^2 - 8m^3\right)\left(6n^2 + 8m^3\right)\).
