Class 9 Math Exercise 1.5 | Question 5

Question 5: 5. Rationalise the denominators of the following:
(i) \\frac{1}{√7})
(ii) \\frac{1}{√7 - √6})
(iii) \\frac{1}{√5 + √2})
(iv) \\frac{1}{√7 - 2})

Explanatory Answer | Exercise 1.5 Question 5

Rationalize The Denominator: (i) \\frac{1}{√7})

The denominator can be rationalised by multiplying the fraction by \\frac{√7}{√7})
\\frac{1}{√7}) = \\frac{1}{√7}) × \\frac{√7}{√7})
= \\frac{√7}{7})

Rationalize The Denominator: (ii) \\frac{1}{√7 - √6})

The denominator can be rationalised by multiplying and dividing the fraction by the conjugate of the denominator, (√7 + √6)
i.e., by multiplying the fraction by \\frac{√7 + √6}{√7 + √6})
\\frac{1}{√7 - √6}) = \\frac{1}{√7 - √6}) × \\frac{√7 + √6}{√7 + √6})

= \\frac{√7 + √6}{\left(√7\right)^2 - \left(√6\right)^2})

= \\frac{√7 + √6}{7 - 6}) = √7 + √6

Rationalize The Denominator: (iii) \\frac{1}{√5 + √2})

The denominator can be rationalised by multiplying and dividing the fraction by the conjugate of the denominator, (√5 - √2)
i.e., by multiplying the fraction by \\frac{√5 - √2}{√5 - √2})
\\frac{1}{√5 + √2}) = \\frac{1}{√5 + √2}) × \\frac{√5 - √2}{√5 - √2})

= \\frac{√5 - √2}{\left(√5\right)^2 - \left(√2\right)^2})

= \\frac{√5 - √2}{5 - 2}) = \\frac{√5 - √2}{3})

Rationalize The Denominator: (iv) \\frac{1}{√7 - 2})

The denominator can be rationalised by multiplying and dividing the fraction by the conjugate of the denominator, (√7 + 2)
i.e., by multiplying the fraction by \\frac{√7 + 2}{√7 + 2})
\\frac{1}{√7 - 2}) = \\frac{1}{√7 - 2}) × \\frac{√7 + 2}{√7 + 2})

= \\frac{√7 + 2}{\left(√7\right)^2 - \left(2\right)^2})

= \\frac{√7 + 2}{7 - 4}) = \\frac{√7 + 2}{3})