Question

please explain solution

Answer 1

Here ,
√3 + ³ √5
Let x =√3
y = ³√5

x^6 = (√2)^6 = (2)³ = 8
y^6 = (³√5)^6 = (5)² = 25
we know , for rationalize we have require (x + y) .

convert x^6 + y^6 to in such a way that here (x + y) must include.

x^6 + y^6 = x^6 +x^5y - x^5y - x^4y² + x⁴y² + x³y³ - x³y³ - x²y⁴ + x²y⁴ + xy^5 - xy^5 - y^6

= x^5( x + y) - x^4y( x + y) + x^3y²( x + y) -x²y³( x + y) + xy⁴(x + y) - y^5(x + y)

= (x + y) { x^5 -x⁴y + x³y² -x²y³ + xy⁴ - y^5 }

now,
x^6 + y^6 = 8 + 25 = 33
( x + y) { x^5 -x⁴y + x³y² - x²y³ + xy⁴ - y^5} = 33

put x = (2)½ and y = (5)⅓ in { x^5 - x⁴y+ x³y² - x²y³ + xy⁴ - y^5 }
and then , 33 divide by it .
then, you got ( x + y)
here , (x + y) is rationalize of √3 + ³√5

hence, {x^5 - x⁴y + x³y² -x²y³ + xy⁴ - y^5 } is rationalise factor where x = (2)½ and y = (5)⅓