Rationalize the denominator of the expression (3√2 − 4√3) / (4√3 − 2√2).

Question

ScanSolve AI Tutor · Accepted Answer

To rationalize the expression, we multiply the numerator and the denominator by the conjugate of the denominator. The denominator is $4\sqrt{3} - 2\sqrt{2}$ , so its conjugate is $4\sqrt{3} + 2\sqrt{2}$ .

Multiply the expression by the conjugate.
$\frac{3\sqrt{2} - 4\sqrt{3}}{4\sqrt{3} - 2\sqrt{2}} \times \frac{4\sqrt{3} + 2\sqrt{2}}{4\sqrt{3} + 2\sqrt{2}}$
Expand the numerator.
$(3\sqrt{2} - 4\sqrt{3})(4\sqrt{3} + 2\sqrt{2})$ $(3\sqrt{2})(4\sqrt{3}) + (3\sqrt{2})(2\sqrt{2}) - (4\sqrt{3})(4\sqrt{3}) - (4\sqrt{3})(2\sqrt{2})$ $12\sqrt{6} + 6(\sqrt{2})^2 - 16(\sqrt{3})^2 - 8\sqrt{6}$ $12\sqrt{6} + 6(2) - 16(3) - 8\sqrt{6}$ $12\sqrt{6} + 12 - 48 - 8\sqrt{6}$
Combine like terms:
$(12 - 48) + (12\sqrt{6} - 8\sqrt{6})$ $-36 + 4\sqrt{6}$
Expand the denominator.
This is in the form $(a-b)(a+b) = a^2 - b^2$ .
$(4\sqrt{3} - 2\sqrt{2})(4\sqrt{3} + 2\sqrt{2}) = (4\sqrt{3})^2 - (2\sqrt{2})^2$ $(16 \times 3) - (4 \times 2)$ $48 - 8$ $40$
Combine the simplified numerator and denominator.
$\frac{-36 + 4\sqrt{6}}{40}$
Simplify the fraction by dividing each term in the numerator by the denominator.
Divide both terms by their greatest common factor, which is 4.
$\frac{-36}{40} + \frac{4\sqrt{6}}{40}$ $-\frac{9}{10} + \frac{\sqrt{6}}{10}$
This can also be written as:
$\frac{\sqrt{6} - 9}{10}$

Rationalize the denominator of the expression (3√2 - 4√3) / (4√3 - 2√2).

Answer

Step-by-step solution

Multiply the expression by the conjugate.

Expand the numerator.

Expand the denominator.

Combine the simplified numerator and denominator.

Simplify the fraction by dividing each term in the numerator by the denominator.

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