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i) x³ + y³ = (x + y)(x² - xy + y² )

Taking the RHS,

(x + y)(x² - xy + y²) = x(x² - xy + y² ) + y(x² - xy + y²)

= x³ - x²y + xy² + x²y - xy² + y³

= x³ + y³ = LHS

ii) x³ - y³ = (x - y)(x² + xy + y²)

Taking the RHS,

(x - y)(x² + xy + y²) = x (x² + xy + y²) - y (x² + xy + y²)

= x³ + x²y + xy² - x²y - xy² - y³

= x³ - y³ = LHS

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

(a + b)(a - b) = a² - b²

(i) 9x² + 6xy + y²

= (3x)² + 2(3x)( y) + (y)²

Identity: (a + b)² = a² + 2ab + b²

Considering a = 3x and b = y 

Hence 9x² + 6xy + y² = (3x + y)²

(ii) 4y² - 4y + 1

= (2y)² - 2(2y)(1) + (1)²

Identity: (a - b)² = a² - 2ab + b²

Considering a = 2y and b = 1

Hence 4y² - 4y + 1 = (2y - 1)²

(iii) x² - y²/100

= x² - (y/10)²

Identity: (a + b)(a - b) = a² - b²

Considering a = x and b = y/10

Hence, x² - y²/100 = (x + y/10)(x - y/10)

i) 4x² + 9y² + 16z² + 12xy - 24yz - 16xz

This can be re-written as:

(2x)² +(3y)² +(-4z)² + 2(2x)(3y) + 2(3y)(-4z) + 2(-4z)(2x) 

This is of the form a² + b² + c² + 2ab + 2bc + 2ca = (a + b + c)²

Here a = 2x, b = 3y, c = -4z

Hence, 4x² + 9y² + 16z² +12xy - 24yz - 16xz = (2x + 3y - 4z)²

ii) 2x² + y² + 8z² - 2√2xy + 4√2yz - 8xz

We know that, 2 = (√2)², √8 = 2√2

Thus, the given expression can be re-written as:

(-√2x)² + (y)² + (2√2z )² + 2(-√2x)(y) + 2(y)(2√2z) + 2(2√2z)(-√2x)

This is of the form a² + b² + c² + 2ab + 2bc + 2ca = (a + b + c)²

Here a = -√2x, b = y, c = 2√2z

Hence 2x² + y² + 8z² - 2√2xy + 4√2yz - 8xz = (-√2x + y + 2√2z)²

Using identity:

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(i) (99)³

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= 

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(ii) (102)³

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(iii) (998)³

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