Rational & Irrational Numbers：練習問題

Which of the following points on a number line best represents the value of \( -\sqrt{10} \)?

If \( c = \sqrt{10} \) and \( d = \sqrt{40} \), which of the following operations results in a rational number?

A rectangle \(ABCD\) has a length of \( \sqrt{125} + \sqrt{5} \) cm and a width of \( \sqrt{80} - \sqrt{20} \) cm. Find the perimeter of the rectangle.

Which of the following integers is the closest to the value of \( \sqrt{130} \) on a number line?

Given that \( a = \sqrt{5} \) and \( b = \sqrt{20} \), which of the following expressions results in a rational number?

Determine whether the value of \( \sqrt{0.81} \) is a rational or an irrational number.

Determine whether the value of the expression \( \sqrt{1.44} - \sqrt[3]{-27} \) is a rational or an irrational number. Show your calculation steps.

If \( x = \sqrt{2} \), evaluate the value of the algebraic fraction \( \frac{x^4 - 4}{x^2 + 2} \) and state whether the result is rational or irrational.

(a) Evaluate the value of \( \sqrt[3]{-125} \).
(b) State whether the value obtained in (a) is a rational number or an irrational number. Justify your answer.
(c) If this value is represented on a number line, identify if it lies between two integers or if it is exactly an integer. If it lies between integers, name them.

In a right-angled triangle, the lengths of the two legs are \( \sqrt{12} \) cm and \( \sqrt{8} \) cm.
(a) Express the lengths of the two legs in the simplest surd form \( a\sqrt{b} \).
(b) Calculate the area of the triangle. Is the result a rational or an irrational number?
(c) Using Pythagoras' Theorem, find the length of the hypotenuse. Express your answer in its simplest surd form.