Rationalize the expression 6sqrt(2) − 2sqrt(10)sqrt(5) - sqrt(3).

Question

ScanSolve AI Tutor · Accepted Answer

To rationalize the expression, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of $\sqrt{5} - \sqrt{3}$ is $\sqrt{5} + \sqrt{3}$ .

Multiply by the conjugate.
$\frac{6\sqrt{2} - 2\sqrt{10}}{\sqrt{5} - \sqrt{3}} \times \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}}$
Expand the numerator.
$(6\sqrt{2} - 2\sqrt{10})(\sqrt{5} + \sqrt{3})$ $(6\sqrt{2})(\sqrt{5}) + (6\sqrt{2})(\sqrt{3}) - (2\sqrt{10})(\sqrt{5}) - (2\sqrt{10})(\sqrt{3})$ $6\sqrt{10} + 6\sqrt{6} - 2\sqrt{50} - 2\sqrt{30}$
Simplify $\sqrt{50}$ : $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$ .
Substitute this back into the expression:
$6\sqrt{10} + 6\sqrt{6} - 2(5\sqrt{2}) - 2\sqrt{30}$ $6\sqrt{10} + 6\sqrt{6} - 10\sqrt{2} - 2\sqrt{30}$
Expand the denominator.
This is in the form $(a-b)(a+b) = a^2 - b^2$ .
$(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3}) = (\sqrt{5})^2 - (\sqrt{3})^2$ $5 - 3$ $2$
Combine the simplified numerator and denominator.
$\frac{6\sqrt{10} + 6\sqrt{6} - 10\sqrt{2} - 2\sqrt{30}}{2}$
Simplify the fraction by dividing each term in the numerator by the denominator.
$\frac{6\sqrt{10}}{2} + \frac{6\sqrt{6}}{2} - \frac{10\sqrt{2}}{2} - \frac{2\sqrt{30}}{2}$ $3\sqrt{10} + 3\sqrt{6} - 5\sqrt{2} - \sqrt{30}$

Rationalize the expression 6sqrt(2) - 2sqrt(10)sqrt(5) - sqrt(3).

Answer

Step-by-step solution

Multiply 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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