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if x = $\sf{\dfrac{1}{ 7 + 4\sqrt{3}}}$ and y = $\sf{\dfrac{1}{ 7 - 4\sqrt{3}}}$ then find the value of x³ + y³.​

Answer 1

★ Given -:

$x = \dfrac{1}{7+4\sqrt{3}}$

$y = \dfrac{1}{7 - 4\sqrt{3}}$

★ To find -:

Value of x³ + y³

★ Solution -:

Refer the attachment

★ Steps followed -:

» Rationalise both the terms

» put the values of x and y in the identity ..

Answer 2

Question : -

If x = (1)/(7+4√3) & y = (1)/(7-4√3) then find the value of x³ + y³ ?

ANSWER

Given : -

x = (1)/(7+4√3) & y = (1)/(7-4√3)

Required to find : -

• value of x³ + y³

Formula used : -

• x³ + y³ = (x+y) (x²-xy+y²)

Solution : -

x = (1)/(7+4√3) & y = (1)/(7-4√3)

We need to find the value of x³ + y³ ?

So,

Value of x = (1)/(7+4√3)

Here,

Let's rationalize the denominator .

Rationalising factor of 7+4√3 = 7-4√3

Multiplying the numerator & denominator with the rationalising factor .

(1)/(7+4√3) x (7-4√3)/(7-4√3)

(7-4√3)/([7+4√3] [7-4√3])

(7-4√3)/([7]²-[4√3]²)

(7-4√3)/(49-16 x 3)

(7-4√3)/(49-48)

(7-4√3)/(1)

(7-4√3)

Similarly,

Value of y = (1)/(7-4√3)

Here,

Let's rationalize the denominator.

Rationalising factor of 7-4√3 = 7+4√3

Multiplying the numerator & denominator with the rationalising factor .

(1)/(7-4√3) x (7+4√3)/(7+4√3)

(7+4√3)/([7-4√3] [7+4√3])

(7+4√3)/([7]²-[4√3]²)

(7+4√3)/(49-48)

(7+4√3)/(1)

(7+4√3)

Hence,

• Value of x = 7-4√3
• Value of y = 7+4√3

For the next calculations let's us these values only .

Now,

Let's find the value of x² & y²

This implies;

x² = (7-4√3)²

This is in the form of;

• (x-y)² = x²+y²-2xy

=> (7)²+(4√3)²-2(7)(4√3)

=> 49+16 x 3-48√3

=> 49+48-48√3

=> 97-48√3

Similarly,

y² = (7+4√3)²

=> (7)²+(4√3)²+2(7)(4√3)

=> 49+16 x 3+48√3

=> 49+48+48√3

=> 97+48√3

So,

• Value of x² = 97-48√3
• Value of y² = 97+48√3

Now,

Let's find the value of x³+y³

Using the formula;

• x³ + y³ = (x+y) (x²-xy+y²)

This implies;

(7-4√3+[7+4√3]) ([97-48√3]-[7-4√3][7+4√3]+[97+48√3])

(7-4√3+7+4√3)([97-48√3]-[{7}²-{4√3}²]+[97+58√3])

(7+7)(97-48√3-[49-48]+97+58√3)

(14)(97-48√3-1+97+58√3)

(14)(97+97-1)

(14)(194-1)

(14)(193)

2702

Therefore,

Value of x³+y³ = 2,702