Question

x^16-y^4 expand the following

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Answer 1

Identity used :

a² - b² = (a +b) (a - b)

Solution :

x¹⁶ - y⁴

= (x⁸)² - (y²)²

= (x⁸ + y² ) (x⁸ - y²)

= (x⁸ + y² ) [(x⁴)² - (y)²]

= (x⁸ + y² ) (x⁴ + y) (x⁴ -y)

Verification :

(x⁸ + y²)(x⁴ + y) (x⁴ - y)

= (x⁸ + y²) [ x⁴ (x⁴ - y) + y(x⁴ - y)]

= (x⁸ + y² ) [ x⁸ - x⁴y + x⁴y - y²]

= (x⁸ + y²) ( x⁸ - y²)

= x⁸ (x⁸ - y²) + y² (x⁸ - y²)

= x¹⁶ - x⁸y² + x⁸y² - y⁴

= x¹⁶ - y⁴

Hence verified

Some other identities :

1. ( a+ b)² = a² + b² + 2ab
2. (a - b)² = a² + b² - 2ab
3. (a + b + c)² = a² + b² + c² + 2ab +2bc + 2ca