Simplify the expression: (sqrt(3) + sqrt(2)) / (3sqrt(2) + 2sqrt(3))

Question

ScanSolve AI Tutor · Accepted Answer

Step 1: Let x = √[3]2 . Note that √[3]4 = x² , since 4 = 2² . √[3]2 + √[3]43√[3]2 + 2√[3]4 = (x + x²)/(3x + 2x²) Step 2: Factor x from the numerator and denominator. Numerator: x + x² = x(1 + x); Denominator: 3x + 2x² = x(3 + 2x); (x(1 + x))/(x(3 + 2x)) = (1 + x)/(3 + 2x) Step 3: Rationalize the denominator by multiplying the numerator and denominator by 9 − 6x + 4x² , using the identity (3 + 2x)(9 − 6x + 4x²) = 43. (1 + x)/(3 + 2x) = ((1 + x)(9 − 6x + 4x²))/(43) Step 4: Expand the numerator. (1 + x)(9 − 6x + 4x²) = 1 · 9 + 1 · (−6x) + 1 · 4x² + x · 9 + x · (−6x) + x · 4x²; = 9 − 6x + 4x² + 9x − 6x² + 4x³; Step 5: Substitute x³ = 2 , so 4x³ = 8 . Combine like terms: 9 + 8 + (−6x + 9x) + (4x² − 6x²) = 17 + 3x − 2x²; Step 6: Substitute back x = √[3]2 and x² = √[3]4 . 17 + 3√[3]2 − 2√[3]443 17 + 3√[3]2 - 2√[3]443

Simplify the expression: (sqrt(3) + sqrt(2)) / (3sqrt(2) + 2sqrt(3))

Answer

Step-by-step solution

Let $x = \sqrt[3]{2}$ .

Factor $ x $ from the numerator and denominator.

Rationalize the denominator by multiplying the numerator and denominator by $9 - 6x + 4x^2$ , using the identity

Expand the numerator.

Substitute $x^3 = 2$ , so $4x^3 = 8$ .

Substitute back $x = \sqrt[3]{2}$ and $x^2 = \sqrt[3]{4}$ .

Try one yourself

Still stuck on this one?

More Mathematics Questions

Simplify the expression: (sqrt(3) + sqrt(2)) / (3*sqrt(2) + 2*sqrt(3))

Answer

Step-by-step solution

Let x=23x = \sqrt[3]{2}x=32​.

Factor $ x $ from the numerator and denominator.

Rationalize the denominator by multiplying the numerator and denominator by 9−6x+4x29 - 6x + 4x^29−6x+4x2, using the identity

Expand the numerator.

Substitute x3=2x^3 = 2x3=2, so 4x3=84x^3 = 84x3=8.

Substitute back x=23x = \sqrt[3]{2}x=32​ and x2=43x^2 = \sqrt[3]{4}x2=34​.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Simplify the expression: (sqrt(3) + sqrt(2)) / (3sqrt(2) + 2sqrt(3))

Let $x = \sqrt[3]{2}$ .

Rationalize the denominator by multiplying the numerator and denominator by $9 - 6x + 4x^2$ , using the identity

Substitute $x^3 = 2$ , so $4x^3 = 8$ .

Substitute back $x = \sqrt[3]{2}$ and $x^2 = \sqrt[3]{4}$ .