Ratinalise denominator :- $\newline$ $\frac{7+3 \sqrt{5}}{7-3 \sqrt{5}}$

Full solution

Q. Ratinalise denominator :-

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\frac{7+3 \sqrt{5}}{7-3 \sqrt{5}}

Identify Conjugate: Step $1$ : Identify the conjugate of the denominator to rationalize it. $\newline$ Conjugate of $7 - 3\sqrt{5}$ is $7 + 3\sqrt{5}$ . $\newline$ Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ $\frac{7 + 3\sqrt{5}}{7 - 3\sqrt{5}} \times \frac{7 + 3\sqrt{5}}{7 + 3\sqrt{5}}$
Apply Difference of Squares: Step $2$ : Apply the difference of squares formula to the denominator. $\newline$ $(7 - 3\sqrt{5})(7 + 3\sqrt{5}) = 7^2 - (3\sqrt{5})^2$ $\newline$ $= 49 - 45 = 4$
Expand Numerator: Step $3$ : Expand the numerator using distributive property. $\newline$ $(7 + 3\sqrt{5})(7 + 3\sqrt{5}) = 7^2 + 2 \cdot 7 \cdot 3\sqrt{5} + (3\sqrt{5})^2$ $\newline$ $= 49 + 42\sqrt{5} + 45$ $\newline$ $= 94 + 42\sqrt{5}$
Divide Numerator by Denominator: Step $4$ : Divide the expanded numerator by the simplified denominator. $\newline$ $\frac{94 + 42\sqrt{5}}{4}$ $\newline$ $= 23.5 + 10.5\sqrt{5}$