Exercise 2 – Animated Solutions ഹരണം
Five questions that use division of square roots, rationalising a denominator and the identity (x + y)(x − y) = x² − y² – plus one beautiful geometry puzzle. Try each question in your notebook first, use the hint if you are stuck, then press Next ▶ to watch the solution grow.
- Identity: (x + y)(x − y) = x² − y², true for irrational numbers too (page 7 of this guide).
- Division rules: √x/y = √x ÷ √y and 1/√x = √x/x (page 7).
- Equilateral triangle: height = √3 × (half the side) (page 4 and page 9).
- Simplifying: √(k² × m) = k√m (page 5).
Question 1 · (√2 + 1)(√2 − 1) = 1 and what it is good for
Use (x + y)(x − y) = x² − y² with x = √2, y = 1. For (i), multiply the top and bottom of the fraction by (√2 + 1) – the denominator becomes 1.
Dividing by 0.414… by hand is painful. Dividing by 1 is trivial. Multiplying top and bottom by the "partner" (√2 + 1) turns the awkward denominator √2 − 1 into the whole number 1. This is called rationalising the denominator – you will use it again and again in Class 10.
Question 2 · An equilateral triangle with height 4 cm
Call the side a. The height cuts the triangle into two right triangles with base a/2. Use Pythagoras, or the shortcut height = √3 × (a/2).
Question 3 · The pinwheel of equilateral triangles
The outer square is four identical rectangles arranged around the small square. In one rectangle, the diagonals meet at its centre, so a red triangle has base = short side and height = half the long side. Equilateral ⇒ height = (√3/2) × base.
Drag the slider. The red triangles are equilateral only when the long side is √3 ≈ 1.732 times the short side – then the apex angle is exactly 60° and all three sides of the red triangle are equal. The ratio outer : inner = (r + 1)/(r − 1) changes with the shape.
Question 4 · Square roots of mixed numbers
Turn the mixed number into an improper fraction (2⅔ = 8/3) and look for a perfect-square factor: 8/3 = 4 × 2/3.
For every whole number n ≥ 2, the mixed number "n and n over (n² − 1)" has a square root that is n times the square root of its fraction part. The decimal check in the last line shows both sides agree.
Question 5 · Which quotients are whole numbers or fractions?
√a ÷ √b = √(a ÷ b). Simplify the fraction a/b first; the answer is a whole number or fraction exactly when both the numerator and the denominator of the simplified fraction are perfect squares.
Try a = 125, b = 50 (irrational), a = 18, b = 8 (fraction 3/2), a = 20, b = 5 (whole number).
Check yourself
- (√a + b)(√a − b) = a − b² – a product of two irrationals can be a whole number. Use it to rationalise: 1/(√2 − 1) = √2 + 1.
- Equilateral triangle: height = √3 × half the side, so side = 2 × height ÷ √3 (then rationalise).
- Geometry puzzles become algebra once you name a length (h) and use √3 for the equilateral height.
- Square roots of fractions: pull out perfect-square factors, √(4 × ⅔) = 2√⅔; quotients √a/√b = √(a/b) are rational only when the simplified fraction has perfect-square top and bottom.