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ConceptOperations on Real Numbers

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#### Question

Rationalise the denominators of the following :-

`(i) 1/sqrt7`

`(ii) 1/(sqrt7-sqrt6)`

`(iii) 1/(sqrt5+sqrt2)`

`(iv) 1/(sqrt7-2)`

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#### Similar questions

Classify the following numbers as rational or irrational :-

`(i) 2-sqrt5`

`(ii) (3+sqrt23)-sqrt23`

`(iii) (2sqrt7)/(7sqrt7)`

`(iv) 1/sqrt2`

`(v) 2π`

Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, π = c/d. This seems to contradict the fact that p is irrational. How will you resolve this contradiction?

Represent `sqrt9.3` on the number line.

Simplify each of the following expressions :-

`(i) (3+sqrt3)(2+sqrt2)`

`(ii) (3+sqrt3)(3-sqrt3)`

`(iii) (sqrt5+sqrt2)^2`

`(iv) (sqrt5-sqrt2)(sqrt5+sqrt2)`