Rationalize the denominator of the expression (2)/(sqrt(5)−7) and leave the answer in surd form.

Question

ScanSolve AI Tutor · Accepted Answer

To leave the answer in surd form, we need to rationalize the denominator.

Write the given expression.
$\frac{2}{\sqrt{5}-\sqrt{7}}$
Multiply the numerator and the denominator by the conjugate of the denominator.
The conjugate of $\sqrt{5}-\sqrt{7}$ is $\sqrt{5}+\sqrt{7}$ .
$\frac{2}{\sqrt{5}-\sqrt{7}} \times \frac{\sqrt{5}+\sqrt{7}}{\sqrt{5}+\sqrt{7}}$
Perform the multiplication.
Use the difference of squares formula $(a-b)(a+b) = a^2 - b^2$ for the denominator.
$\frac{2(\sqrt{5}+\sqrt{7})}{(\sqrt{5}-\sqrt{7})(\sqrt{5}+\sqrt{7})} = \frac{2(\sqrt{5}+\sqrt{7})}{(\sqrt{5})^2 - (\sqrt{7})^2}$
Simplify the denominator.
$\frac{2(\sqrt{5}+\sqrt{7})}{5 - 7} = \frac{2(\sqrt{5}+\sqrt{7})}{-2}$
Simplify the fraction by dividing the numerator by the denominator.
$\frac{2(\sqrt{5}+\sqrt{7})}{-2} = -(\sqrt{5}+\sqrt{7})$
Distribute the negative sign.
$-(\sqrt{5}+\sqrt{7}) = -\sqrt{5}-\sqrt{7}$
The simplified expression in surd form is $-\sqrt{5}-\sqrt{7}$ .

Rationalize the denominator of the expression (2)/(sqrt(5)-7) and leave the answer in surd form.