Ratinalise denominator :- (7+3sqrt5)/(7-3sqrt5)

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

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Math Problems

Algebra 1

Simplify radical expressions using conjugates

Full solution

Q. Ratinalise denominator :-

\newline

\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. $\newline$ $\frac{94 + 42\sqrt{5}}{4}$ $\newline$ $= 23.5 + 10.5\sqrt{5}$