Question

rationalize the denominator in the given expression and present your answer simplified. use exact values. do not leave a negative sign in the denominator.\\(\frac{sqrt{11} + 5}{sqrt{11} - 2} = \square\\)

Explanation:

Step1: Multiply by conjugate of denominator

Multiply numerator and denominator by $\sqrt{11}+2$:
$$\frac{\sqrt{11}+5}{\sqrt{11}-2} \times \frac{\sqrt{11}+2}{\sqrt{11}+2} = \frac{(\sqrt{11}+5)(\sqrt{11}+2)}{(\sqrt{11}-2)(\sqrt{11}+2)}$$

Step2: Expand denominator (difference of squares)

Calculate denominator using $(a-b)(a+b)=a^2-b^2$:
$$(\sqrt{11})^2 - 2^2 = 11 - 4 = 7$$

Step3: Expand numerator (FOIL method)

Multiply terms in the numerator:
$$(\sqrt{11})(\sqrt{11}) + (\sqrt{11})(2) + 5(\sqrt{11}) + 5(2) = 11 + 2\sqrt{11} + 5\sqrt{11} + 10$$

Step4: Simplify numerator

Combine like terms and constants:
$$11+10 + (2\sqrt{11}+5\sqrt{11}) = 21 + 7\sqrt{11}$$

Step5: Simplify the fraction

Factor and cancel common terms:
$$\frac{21 + 7\sqrt{11}}{7} = \frac{7(3 + \sqrt{11})}{7} = 3 + \sqrt{11}$$

Answer:

$3 + \sqrt{11}$