Questions: Rationalize the denominator. Write your answer in exact simplified form. sqrt(48) / (sqrt(26) + 5) =

Solution Steps

To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(\sqrt{26} + 5\) is \(\sqrt{26} - 5\). This will eliminate the square root in the denominator.

To rationalize the denominator of \(\frac{\sqrt{48}}{\sqrt{26} + 5}\), we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(\sqrt{26} + 5\) is \(\sqrt{26} - 5\).

We multiply the numerator and the denominator by \(\sqrt{26} - 5\): \[ \frac{\sqrt{48}}{\sqrt{26} + 5} \cdot \frac{\sqrt{26} - 5}{\sqrt{26} - 5} = \frac{\sqrt{48}(\sqrt{26} - 5)}{(\sqrt{26} + 5)(\sqrt{26} - 5)} \]

Simplify the numerator: \[ \sqrt{48}(\sqrt{26} - 5) = 4\sqrt{3}(\sqrt{26} - 5) = 4\sqrt{3}\sqrt{26} - 20\sqrt{3} = 4\sqrt{78} - 20\sqrt{3} \]

Simplify the denominator using the difference of squares: \[ (\sqrt{26} + 5)(\sqrt{26} - 5) = (\sqrt{26})^2 - 5^2 = 26 - 25 = 1 \]

Combine the simplified numerator and denominator: \[ \frac{4\sqrt{78} - 20\sqrt{3}}{1} = 4\sqrt{78} - 20\sqrt{3} \]

Final Answer