The rational numbers satisfy the commutative, associative and distributive laws for addition and multiplication Moreover, if we add, subtract, multiply or divide (except by zero) two rational numbers, we still get a rational number (that is, rational numbers are ‘closed’ with respect to addition, subtraction, multiplication and division). It turns out that irrational numbers also satisfy the commutative, associative and distributive laws for addition and multiplication. However, the sum, difference, quotients and products of irrational numbers are not always irrational.

√5 + (- √5) = 0

√3 - √3= 0

(√2)(√2)= 2

(√7)/( √7)= 1

All are rational numbers.

When we add and multiply a rational number with an irrational number.

For example, √6 is an irrational number so when we add or subtract any rational number to an irrational number the result will be irrational number only.

Example :

-√7 and 3√7 -------------> Like terms.

2√3 and √5 --------------> Unlike terms.

2√11 and 5√6 ------------> Unlike terms.

1) Add the 2√2 + 5√3 and √2 - 3√3

2√2 + 5√3 + √2 + (- 3√3)

=

= 3 √2 + 2√3

-------------------------------------------------------------

2) Add the -6√3 + 3√2 and -2√2 – 4 √3

-6√3 + 3√2 + (-2√2 – 4 √3)

=

= -10√3 + 1√2

-------------------------------------------------------------

3) Multiply (2√3) and (-3√5)

2 √3 x (-3√5)

= (2 x -3) √3 x √5

= - 6 √(3x5)

= -6 √15

-------------------------------------------------------------

4) Multiply 3√2 ( 2 + 4√3)

3√2 ( 2 + 4√3)

= 3√2 x 2 + 3√2 x 4√3 [ use a distributive law]

= (3x2)√2 + (3x4)( √2 x √3)

= 6√2 + 12√6

-------------------------------------------------------------

5) Divide √15 by √3

√15 / √3

= (√3 x √5) / √3

= √5

• Real Numbers

• Representation of real-numbers on number line

• Operations on Real Numbers

• Rationalization of denominator

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