Operations on Real Numbers

Here we will be learning operations on Real Numbers.

Operations on Real Numbers Rules

• The sum or difference of a rational number and an irrational number is irrational.

• The product or quotient of a non-zero rational number with an irrational number is irrational number.
• When two irrational numbers are added, subtracted, multiplied or divided, the result may be a rational or an irrational number.

If a and b are positive real numbers, then we have,

• ${\sqrt {ab}}={\sqrt {a}}{\sqrt {b}}$
• ${\sqrt {\frac {a}{b}}}={\frac {\sqrt {a}}{\sqrt {b}}}$
• $({\sqrt {a}}+{\sqrt {b}})({\sqrt {a}}-{\sqrt {b}})=a-b$

• $(a+{\sqrt {b}})(a-{\sqrt {b}})=a^{2}-b$
• $({\sqrt {a}}+{\sqrt {b}})({\sqrt {c}}+{\sqrt {d}})={\sqrt {ac}}+{\sqrt {ad}}+{\sqrt {bc}}+{\sqrt {bd}}$
• $({\sqrt {a}}+{\sqrt {b}})^{2}=a+2{\sqrt {ab}}+b$

Examples

1.$({\sqrt {11}}+{\sqrt {7}})({\sqrt {11}}-{\sqrt {b}})$

$11-7=4$

2.$({\sqrt {3}}+{\sqrt {7}})^{2}$

$({\sqrt {3}})^{2}+2({\sqrt {3}})({\sqrt {7}})+({\sqrt {7}})^{2}$

$3+2({\sqrt {21}})+7$

$10+2({\sqrt {21}})$

3. $(5+{\sqrt {7}})(2+{\sqrt {5}})$

$10+5{\sqrt {5}}+2{\sqrt {7}}+{\sqrt {35}}$

4.$({\sqrt {7}}+{\sqrt {5}})({\sqrt {7}}-{\sqrt {5}})$

$({\sqrt {7}})^{2}-({\sqrt {5}})^{2}$

$7-5=2$