Show that the square of $$\dfrac{(\sqrt{26 - 15\sqrt{3}})}{(5\sqrt{2} - \sqrt{38 + 5\sqrt{3}})}$$ is a rational number.
Given,
$$ x = \dfrac{\sqrt{26 - 15\sqrt{3}}}{5\sqrt{2} - \sqrt{38 + 5\sqrt{3}}}$$
$$x^2 = \dfrac{26 - 15\sqrt{3}}{50 + 38 + 5\sqrt{3} - 10\sqrt{76 + 10\sqrt{3}}}$$
$$\Rightarrow x^2 = \dfrac{26 - 15\sqrt{3}}{88 + 5\sqrt{3} - 10\sqrt{75 + 1 + 10\sqrt{3}}}$$
$$\Rightarrow x^2 = \dfrac{26 - 15\sqrt{3}}{88 + 5\sqrt{3} - 10\sqrt{(5\sqrt{3} + 1)^2}}$$
$$\Rightarrow x^2 = \dfrac{26 - 15\sqrt{3}}{88 + 5\sqrt{3} - 10( 5\sqrt{3} + 1)}$$
$$\Rightarrow x^2 = \dfrac{26 - 15\sqrt{3}}{3(26 - 15\sqrt{3})} = \dfrac{1}{3}$$ which is a rational number.
State whether the following statements are true or false. Justify your answers. (i) Every irrational number is a real number. (ii) Every point on the number line is of the form $$\sqrt{m}$$, where $$m$$ is a natural number. (iii) Every real number is an irrational number.
Classify the following numbers as rational or irrational: (i) $$\sqrt{23}$$ (ii) $$\sqrt{225}$$ (iii) $$0.3796$$ (iv) $$7.478478...$$ (v) $$1.101001000100001...$$
Recall, $$\pi$$ is defined as the ratio of the circumference(say c) of a circle to its diameter(say d). That is, $$\pi=\displaystyle\frac{c}{d}$$. This seems to contradict the fact that $$\pi$$ is irrational. How will you resolve this contradiction?
Prove that $$3+2\surd{5}$$ is irrational.
Prove that $$\sqrt{3}$$ is irrational.
Show that $$3\sqrt{2}$$ is irrational.
The fact that $$3+2\sqrt{5}$$ is irrational is because
Prove that the following are irrational. $$\displaystyle\frac{1}{\sqrt{2}}$$.
Prove that $$6+\sqrt{2}$$ is irrational.
Prove that $$\sqrt{5}-\sqrt{3}$$ is not a rational number.