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

Transcript text: Rationalize the denominator. Write your answer in exact simplified form. \[ \frac{\sqrt{48}}{\sqrt{26}+5}= \] $\square$

Solution

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.

Step 1: Identify the Conjugate of 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 $\sqrt{26} - 5$.

Step 2: Multiply Numerator and Denominator by the Conjugate

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

Step 3: Simplify the Numerator

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

Step 4: Simplify the Denominator

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

Step 5: Combine and Simplify the Expression

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