Check is rational number and irrational number?

Foundation

Mathematics Foundation

Surds

Irrational Number

Rationalisation

Question

Check $\frac{2 \sqrt{45} + 3 \sqrt{20}}{2 \sqrt{5}}$ is rational number and irrational number?

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$\frac{2 \sqrt{45} + 3 \sqrt{20}}{2 \sqrt{5}}$

= \frac{2 \sqrt{(3.3) . 5} + 3 \sqrt{(2.2) . 5}}{2 \sqrt{5}}

= \frac{2.3 \sqrt{5} + 3.2 \sqrt{5}}{2 \sqrt{5}}

= \frac{6 \sqrt{5} + 6 \sqrt{5}}{2 \sqrt{5}}

= \frac{12 \sqrt{5}}{2 \sqrt{5}}

= 6

Hence, it is a rational number.