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Date

15//4//2024
Q1) Simplify by rationalising the denominator:
(a)
(5+sqrt6)/(5-sqrt6)
(b)
(7+3sqrt5)/(7-3sqrt5)

Date $\newline$ $15 / 4 / 2024$ $\newline$ Q $1$ ) Simplify by rationalising the denominator: $\newline$ (a) $\frac{5+\sqrt{6}}{5-\sqrt{6}}$ $\newline$ (b) $\frac{7+3 \sqrt{5}}{7-3 \sqrt{5}}$

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Simplify rational expressions

Full solution

Q. Date

\newline

15 / 4 / 2024

\newline

Q

1

) Simplify by rationalising the denominator:

\newline

(a)

\frac{5+\sqrt{6}}{5-\sqrt{6}}

\newline

(b)

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

Multiply by Conjugate: For $(a) \frac{5+\sqrt{6}}{5-\sqrt{6}}$ , multiply both numerator and denominator by the conjugate of the denominator, which is $(5+\sqrt{6})$ . $\frac{5+\sqrt{6}}{5-\sqrt{6}} \times \frac{5+\sqrt{6}}{5+\sqrt{6}}$
Numerator Multiplication: Now, perform the multiplication in the numerator: $(5+\sqrt{6})\cdot(5+\sqrt{6}) = 5^2 + 2\cdot 5\cdot\sqrt{6} + (\sqrt{6})^2$ .
Denominator Multiplication: Perform the multiplication in the denominator: $(5-\sqrt{6})\cdot(5+\sqrt{6}) = 5^2 - (\sqrt{6})^2$ .
Calculate Squares: Calculate the squares and simplify: Numerator = $25 + 10\sqrt{6} + 6$ , Denominator = $25 - 6$ .
Combine Like Terms: Combine like terms: Numerator = $31 + 10\sqrt{6}$ , Denominator = $19$ .
Simplified Expression: Write the simplified expression for $(a): \frac{31 + 10\sqrt{6}}{19}$ .
Multiply by Conjugate: For $(b) \frac{7+3\sqrt{5}}{7-3\sqrt{5}}$ , multiply both numerator and denominator by the conjugate of the denominator, which is $(7+3\sqrt{5})$ . $\frac{7+3\sqrt{5}}{7-3\sqrt{5}} \cdot \frac{7+3\sqrt{5}}{7+3\sqrt{5}}$
Numerator Multiplication: Now, perform the multiplication in the numerator: $(7+3\sqrt{5})\times(7+3\sqrt{5}) = 7^2 + 2\times 7\times 3\sqrt{5} + (3\sqrt{5})^2$ .
Denominator Multiplication: Perform the multiplication in the denominator: $(7-3\sqrt{5})\cdot(7+3\sqrt{5}) = 7^2 - (3\sqrt{5})^2$ .
Calculate Squares: Calculate the squares and simplify: Numerator = $49 + 42\sqrt{5} + 45$ , Denominator = $49 - 45$ .
Combine Like Terms: Combine like terms: Numerator = $94 + 42\sqrt{5}$ , Denominator = $4$ .
Simplified Expression: Write the simplified expression for $(b)$ : $\frac{94 + 42\sqrt{5}}{4}$ .
Divide Numerator by Denominator: Now, simplify the expression for $b$ by dividing each term in the numerator by the denominator: $\frac{94}{4}$ + $\frac{42\sqrt{5}}{4}$ .

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