Find Whether Variables Represent Rational or Irrational Numbers
Question: In the following equations, find whether the variables represent rational or irrational numbers:
- (i) \( x^2 = 5 \)
- (ii) \( y^2 = 9 \)
- (iii) \( z^2 = 0.04 \)
- (iv) \( u^2 = \frac{17}{4} \)
- (v) \( v^2 = 3 \)
- (vi) \( w^2 = 27 \)
- (vii) \( t^2 = 0.4 \)
Concept Used:
If a number is a perfect square, its square root is rational. Otherwise, it is irrational. :contentReference[oaicite:0]{index=0}
Solution:
(i) \( x^2 = 5 \)
\[ x = \pm \sqrt{5} \]
5 is not a perfect square ⇒ \( \sqrt{5} \) is irrational.
Conclusion: x is irrational
(ii) \( y^2 = 9 \)
\[ y = \pm 3 \]
3 is rational.
Conclusion: y is rational
(iii) \( z^2 = 0.04 \)
\[ z = \pm \sqrt{0.04} = \pm 0.2 \]
0.2 is a terminating decimal ⇒ rational.
Conclusion: z is rational
(iv) \( u^2 = \frac{17}{4} \)
\[ u = \pm \frac{\sqrt{17}}{2} \]
17 is not a perfect square ⇒ irrational.
Conclusion: u is irrational
(v) \( v^2 = 3 \)
\[ v = \pm \sqrt{3} \]
\( \sqrt{3} \) is irrational.
Conclusion: v is irrational
(vi) \( w^2 = 27 \)
\[ w = \pm \sqrt{27} = \pm 3\sqrt{3} \]
\( \sqrt{3} \) is irrational ⇒ whole expression is irrational.
Conclusion: w is irrational
(vii) \( t^2 = 0.4 \)
\[ t = \pm \sqrt{0.4} = \pm \sqrt{\frac{2}{5}} \]
\( \sqrt{2} \) is irrational ⇒ value is irrational.
Conclusion: t is irrational
Final Answers:
- x → Irrational
- y → Rational
- z → Rational
- u → Irrational
- v → Irrational
- w → Irrational
- t → Irrational
Key Points:
- Square root of a perfect square → Rational
- Square root of a non-perfect square → Irrational
- Multiplying/dividing irrational numbers keeps them irrational (generally)