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

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

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Answer

17+323−24343\dfrac{17 + 3\sqrt[3]{2} - 2\sqrt[3]{4}}{43}

Step 1: Let x=23x = \sqrt[3]{2}. Note that 43=x2\sqrt[3]{4} = x^2, since 4=224 = 2^2.

23+43323+243=x+x23x+2x2\frac{\sqrt[3]{2} + \sqrt[3]{4}}{3\sqrt[3]{2} + 2\sqrt[3]{4}} = \frac{x + x^2}{3x + 2x^2}

Step 2: Factor xx from the numerator and denominator.

Numerator: x+x2=x(1+x)x + x^2 = x(1 + x)

Denominator: 3x+2x2=x(3+2x)3x + 2x^2 = x(3 + 2x)

x(1+x)x(3+2x)=1+x3+2x\frac{x(1 + x)}{x(3 + 2x)} = \frac{1 + x}{3 + 2x}

Step 3: Rationalize the denominator by multiplying the numerator and denominator by 9−6x+4x29 - 6x + 4x^2, using the identity

(3+2x)(9−6x+4x2)=43.(3 + 2x)(9 - 6x + 4x^2) = 43. 1+x3+2x=(1+x)(9−6x+4x2)43\frac{1 + x}{3 + 2x} = \frac{(1 + x)(9 - 6x + 4x^2)}{43}

Step 4: Expand the numerator.

(1+x)(9−6x+4x2)=1⋅9+1⋅(−6x)+1⋅4x2+x⋅9+x⋅(−6x)+x⋅4x2(1 + x)(9 - 6x + 4x^2) = 1 \cdot 9 + 1 \cdot (-6x) + 1 \cdot 4x^2 + x \cdot 9 + x \cdot (-6x) + x \cdot 4x^2 =9−6x+4x2+9x−6x2+4x3= 9 - 6x + 4x^2 + 9x - 6x^2 + 4x^3

Step 5: Substitute x3=2x^3 = 2, so 4x3=84x^3 = 8. Combine like terms:

9+8+(−6x+9x)+(4x2−6x2)=17+3x−2x29 + 8 + (-6x + 9x) + (4x^2 - 6x^2) = 17 + 3x - 2x^2

Step 6: Substitute back x=23x = \sqrt[3]{2} and x2=43x^2 = \sqrt[3]{4}.

17+323−24343\frac{17 + 3\sqrt[3]{2} - 2\sqrt[3]{4}}{43}

\dfrac{17 + 3\sqrt[3]{2 - 2\sqrt[3]{4}}{43}}

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Quick Answer

Let x = [3]2 . Note that [3]4 = x^2 , since 4 = 2^2 .

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

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI Answer
Step 1: Let x = [3]2 . Note that [3]4 = x^2 , since 4 = 2^2 . [3]2 + [3]43[3]2 + 2[3]4 = (x + x^2)/(3x + 2x^2) Step 2: Factor x from the numerator and denominator. Numerator: x + x^2 = x(1 + x) Denominator: 3x + 2x^2 = 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 + 4x^2 , using the identity (3 + 2x)(9 - 6x + 4x^2) = 43. (1 + x)/(3 + 2x) = ((1 + x)(9 - 6x + 4x^2))/(43) Step 4: Expand the numerator. (1 + x)(9 - 6x + 4x^2) = 1 · 9 + 1 · (-6x) + 1 · 4x^2 + x · 9 + x · (-6x) + x · 4x^2 = 9 - 6x + 4x^2 + 9x - 6x^2 + 4x^3 Step 5: Substitute x^3 = 2 , so 4x^3 = 8 . Combine like terms: 9 + 8 + (-6x + 9x) + (4x^2 - 6x^2) = 17 + 3x - 2x^2 Step 6: Substitute back x = [3]2 and x^2 = [3]4 . 17 + 3[3]2 - 2[3]443 17 + 3[3]2 - 2[3]443