Identifying the Rational Number

A rational number is defined as any number that can be expressed as a fraction $\frac{p}{q}$, where $p$ (numerator) and $q$ (denominator) are integers, and $q$ is not equal to zero ($q \neq 0$). Numbers that cannot be expressed in this form are irrational.

Analyzing the Options

We need to determine which of the given options simplifies to a rational number.

• Option 1: $(\sqrt{2} + \sqrt{5})^2$
  • Using the algebraic identity $(a+b)^2 = a^2 + 2ab + b^2$:
  • $= (\sqrt{2})^2 + 2(\sqrt{2})(\sqrt{5}) + (\sqrt{5})^2$
  • $= 2 + 2\sqrt{10} + 5$
  • $= 7 + 2\sqrt{10}$
  • Since $\sqrt{10}$ is an irrational number, $7 + 2\sqrt{10}$ is also irrational.
• Option 2: $2 + \sqrt{5}$
  • The square root of 5, $\sqrt{5}$, is an irrational number.
  • Adding an integer (2) to an irrational number always results in an irrational number.
• Option 3: $2 - \sqrt{5}$
  • Similar to Option 2, $\sqrt{5}$ is irrational.
  • Subtracting an irrational number from an integer (2) results in an irrational number.
• Option 4: $(\sqrt{2} + \frac{1}{\sqrt{8}})^2$
  • First, simplify the term $\sqrt{8}$:
  • $\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$
  • Substitute this back into the fraction: $\frac{1}{\sqrt{8}} = \frac{1}{2\sqrt{2}}$
  • Now, consider the expression inside the parentheses: $\sqrt{2} + \frac{1}{2\sqrt{2}}$
  • To add these terms, find a common denominator:
  • $\frac{\sqrt{2}}{1} + \frac{1}{2\sqrt{2}} = \frac{\sqrt{2} \times (2\sqrt{2})}{2\sqrt{2}} + \frac{1}{2\sqrt{2}} = \frac{2 \times 2}{2\sqrt{2}} + \frac{1}{2\sqrt{2}} = \frac{4+1}{2\sqrt{2}} = \frac{5}{2\sqrt{2}}$
  • Now, square this result:
  • $(\frac{5}{2\sqrt{2}})^2 = \frac{5^2}{(2\sqrt{2})^2} = \frac{25}{(2^2)(\sqrt{2})^2} = \frac{25}{4 \times 2} = \frac{25}{8}$
  • The result $\frac{25}{8}$ is a fraction of two integers ($p=25, q=8$), and the denominator is non-zero. Therefore, it is a rational number.

Conclusion

After simplifying each option, Option 4, $(\sqrt{2} + \frac{1}{\sqrt{8}})^2$, is the only expression that results in a rational number ($\frac{25}{8}$).