Forms of Numbers and Number Systems：練習題

Simplify the expression \(\sqrt{50} - \sqrt{18} + \sqrt{8}\) into the form \(k\sqrt{2}\), where \(k\) is an integer.

Evaluate and simplify the expression:
\(\frac{1}{\sqrt{3} + \sqrt{2}} + \frac{1}{\sqrt{4} + \sqrt{3}} + \frac{1}{\sqrt{5} + \sqrt{4}} + \dots + \frac{1}{\sqrt{100} + \sqrt{99}}\)

Express \(\frac{14}{3 - \sqrt{2}}\) in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are integers.

Given that \((3 - 2\sqrt{5})^2 = p + q\sqrt{5}\), where \(p\) and \(q\) are integers, find the value of \(p + q\).

Simplify \(\sqrt{11 - 4\sqrt{7}}\) into the form \(\sqrt{a} - \sqrt{b}\), where \(a\) and \(b\) are positive integers and \(a > b\).

Simplify the expression \(\sqrt{45} - \sqrt{20}\) and express your answer in the form \(\sqrt{a}\), where \(a\) is a positive integer.

The diagram below shows a rectangle. Calculate the exact area of this rectangle in square units.

Given that \(x = \sqrt{7} - \sqrt{5}\), find the exact value of \(x^2 + \frac{4}{x^2}\).

Consider the expression \(A = \sqrt{48} - \sqrt{12}\).
(a) Express \(A\) in the form \(k\sqrt{3}\), where \(k\) is an integer.
(b) Find the value of \(A^2\).

Consider the expression \(E = \frac{\sqrt{48} - \sqrt{12}}{\sqrt{3} + 1}\).

(a) Simplify the numerator \(\sqrt{48} - \sqrt{12}\) into the form \(k\sqrt{3}\), where \(k\) is an integer.
(b) By rationalizing the denominator, simplify the full expression \(E\) into the form \(a - \sqrt{b}\), where \(a\) and \(b\) are integers.