1) Упростите, применив формулы сокращённого умножения: а) (m-n^1/2)^2 + (m+n^1/2)^2; б) (m^1/3+2n^1/2)^2...

1) а) (m-n^1/2)^2 + (m+n^1/2)^2 = m^2 - 2mn^1/2 + n + m^2 + 2mn^1/2 + n = 2m^2 + 2n;

б) (m^1/3+2n^1/2)^2 - (m^1/3-2n^1/2)^2 = m^2/3 + 4mn^1/2 + 4n - (m^2/3 - 4mn^1/2 + 4n) = 8mn^1/2;

в) (m^1/4-n^1/2)*(m^1/4+n^1/2) = m^1/2 - n;

г) (m^1/2+n)(m-m^1/2 n+n^2) = m^3/2 + mn - m^1/2n - n^2.

2) а) (7^1/2-3^1/2)^2 + (7^1/2+3^1/2)^2 = 4 + 16 = 20;

б) ((3^1/4 +27^1/4)^2 -12) ((3^1/4 -27^1/4)^2 +12) = (3^1/2)^2 (3^1/2)^2 = 9 * 9 = 81.

3) а) (x-y) / (x^1/2 - y^1/2) = (x-y) / (x^1/2 - y^1/2) * (x^1/2 + y^1/2) / (x^1/2 + y^1/2) = x^3/2 + y^3/2;

б) (x^3/2+y^3/2) / (x-x^1/2 y^1/2 +y) = (x^3/2+y^3/2) / (x-x^1/2 y^1/2 +y) * (x^1/2 + y^1/2) / (x^1/2 + y^1/2) = x^2 + xy + y^2.

1) Упростите, применив формулы сокращённого умножения:

а) ((m-n^{1/2})^2 + (m+n^{1/2})^2)

Используем формулу (a^2 + b^2 = (a+b)^2 - 2ab), где (a = m - n^{1/2}) и (b = m + n^{1/2}).

[ (m-n^{1/2})^2 + (m+n^{1/2})^2 = [(m-n^{1/2}) + (m+n^{1/2})]^2 - 2(m-n^{1/2})(m+n^{1/2}) ] [ = (2m)^2 - 2(m^2 - (n^{1/2})^2) = 4m^2 - 2m^2 + 2n = 2m^2 + 2n ]

б) ((m^{1/3}+2n^{1/2})^2 - (m^{1/3}-2n^{1/2})^2)

Используем формулу разности квадратов (a^2 - b^2 = (a+b)(a-b)), где (a = m^{1/3}+2n^{1/2}) и (b = m^{1/3}-2n^{1/2}).

[ (m^{1/3}+2n^{1/2})^2 - (m^{1/3}-2n^{1/2})^2 = [(m^{1/3}+2n^{1/2}) + (m^{1/3}-2n^{1/2})][(m^{1/3}+2n^{1/2}) - (m^{1/3}-2n^{1/2})] ] [ = (2m^{1/3})(4n^{1/2}) = 8m^{1/3}n^{1/2} ]

в) ((m^{1/4}-n^{1/2})(m^{1/4}+n^{1/2}))

Это стандартная формула разности квадратов (a^2 - b^2), где (a = m^{1/4}) и (b = n^{1/2}).

[ (m^{1/4}-n^{1/2})(m^{1/4}+n^{1/2}) = (m^{1/4})^2 - (n^{1/2})^2 = m^{1/2} - n ]

г) ((m^{1/2}+n)(m-m^{1/2} \cdot n+n^2))

Раскроем скобки и упростим:

[ (m^{1/2}+n)(m-m^{1/2} \cdot n+n^2) = m^{1/2} \cdot m + m^{1/2} \cdot (-m^{1/2} \cdot n) + m^{1/2} \cdot n^2 + n \cdot m - n \cdot m^{1/2} \cdot n + n \cdot n^2 ] [ = m^{3/2} - m + m^{1/2}n^2 + nm - n^2 + n^3 ]

2) Вычислите:

а) ((7^{1/2}-3^{1/2})^2 + (7^{1/2}+3^{1/2})^2) используя решение из п.1а:

[ (7^{1/2}-3^{1/2})^2 + (7^{1/2}+3^{1/2})^2 = 2 \cdot 7 + 2 \cdot 3 = 14 + 6 = 20 ]

б) (((3^{1/4} +27^{1/4})^2 -12) * ((3^{1/4} -27^{1/4})^2 +12))

Для упрощения, обозначим (a = 3^{1/4}), (b = 27^{1/4} = 3^{3/4}), тогда:

[ ((a + b)^2 -12) ((a - b)^2 +12) = (a^2 + 2ab + b^2 - 12) (a^2 - 2ab + b^2 +12) ] [ = (3 + 2\cdot3 + 9 - 12) (3 - 2\cdot3 + 9 + 12) = (12) (21) = 252 ]

3) Сократите дробь:

а) ((x-y) / (x^{1/2} - y^{1/2}))

Обозначим (a = x^{1/2}) и (b = y^{1/2}), тогда:

[ \frac{x-y}{x^{1/2} - y^{1/2}} = \frac{a^2 - b^2}{a - b} = \frac{(a-b)(a+b)}{a - b} = a + b = x^{1/2} + y^{1/2} ]

б) ((x^{3/2}+y^{3/2}) / (x-x^{1/2} \cdot y^{1/2} +y))

Обозначим (a = x^{1/2}) и (b = y^{1/2}), тогда:

[ \frac{x^{3/2}+y^{3/2}}{x - x^{1/2} \cdot y^{1/2} + y} = \frac{a^3 + b^3}{a^2 - ab + b^2} ] [ = \frac{(a+b)(a^2 - ab + b^2)}{a^2 - ab + b^2} = a + b = x^{1/2} + y^{1/2} ]

1) а) 2m^2 + 2n б) 4m^2 + 4n в) m^(1/2) г) m + n

2) а) 14 б) -3

3) а) x^(1/2) + y^(1/2) б) x^(1/2) - y^(1/2)