Question

Check whether $-\frac{\sqrt{63}}{\sqrt{448}}$ is a rational number or an irrational number.

Answer 1

Irrational numbers are numbers that are neither terminating nor recurring and cannot be expressed as a ratio of integers.

Rational numbers are non-terminating repeating decimals that can be represent in $p/q$, where$q\ne 0$ & $p,q\in Z$.

Given $-\frac{\sqrt{63}}{\sqrt{448}}$

It can be rewritten as:

$$\begin{gathered} -\frac{\sqrt{63}}{\sqrt{448}}=-\frac{\sqrt{7\times 9}}{\sqrt{2^6\times 7}} \\ =-\frac{\sqrt{7\times 3^2}}{\sqrt{2^6\times 7}} \\ =-\frac{3\sqrt{7}}{2^3\sqrt{7}} \\ =-\frac{3}{8} \end{gathered}$$

Which is clearly a rational number as it is in $\frac{p}{q}$ form, where$q\ne 0$ & $p,q\in Z$.

So, $-\frac{\sqrt{63}}{\sqrt{448}}$ is a rational number.