Division ഹരണം
Every multiplication can be read backwards as a division: 2 × 3 = 6 tells us 6 ÷ 2 = 3. In Section 1 we found √2 × √3 = √6. Read backwards, that gives us the rule for dividing square roots – and a neat trick for turning numbers like 1/√2 into easy decimals.
- Key idea 1: √x × √y = √(xy) for any positive numbers x, y (page 5).
- A fraction does not change when its top and bottom are multiplied by the same number: ½ = 2/4 = 3/6.
- √(a²) = a for any positive a, e.g. √4 = 2, √(x²) = x.
- Decimals: √2 ≈ 1.414, √3 ≈ 1.732, √6 ≈ 2.449.
1. Multiplication read backwards
Think of the rectangle with sides √2 and √3 from Section 1. Its area is √6. Divide the area by one side and you must get the other side.
x, y എന്ന ഏതു രണ്ട് അധിസംഖ്യകളെടുത്താലും √x × √y = √z എന്ന ഗുണനത്തെ ഹരണമായി
√z√x = √y √z√y = √xIf √x × √y = √z, then √z ÷ √x = √y and √z ÷ √y = √x. For example √6/√2 = √3 and √6/√3 = √2.
2. Dividing under the root
Here is a second way to look at the same numbers. Since 6 ÷ 2 = 3, we have √(6/2) = √3 – and we just saw that √6/√2 = √3 as well. So
√62 = √6√2 and √63 = √6√3It works with fractions under the root too. By Key idea 1, √3 × √(2/3) = √(3 × 2/3) = √2, and reading that backwards (Key idea 2) gives √(2/3) = √2 ÷ √3:
√23 = √2√3x, y എന്ന ഏതു രണ്ട് അധിസംഖ്യകളെടുത്താലും
√xy = √x√yThe square root of a fraction is the square root of the top divided by the square root of the bottom.
Both ways of computing give the same decimal every time. When x/y is a perfect square (try x = 18, y = 2, or x = 9, y = 4) the quotient of two square roots is a whole number or a simple fraction.
3. Computing √½ – the easy way
How big is √(1/2)? Key idea 3 says √(1/2) = 1/√2 = 1 ÷ 1.414… Dividing by a long decimal is painful. The textbook shows a cleverer route.
Rewrite the fraction so that the bottom becomes a perfect square: ½ = 2/4. Then √(2/4) = √2/√4 = √2/2, and the decimal is simply 1.414 ÷ 2 = 0.707. The same trick gives 1/√3 = √(1/3) = √(3/9) = √3/3 ≈ 1.732 ÷ 3 = 0.577.
4. The general rule: 1/√x = √x / x
x ഏത് അധിസംഖ്യയായാലും
1√x = √1x = √xx² = √x√(x²) = √xxDividing by √x is the same as multiplying by √x and dividing by x. Writing a fraction so that its bottom has no square root is called rationalising the denominator.
The "hard way" divides 1 by a decimal; the "easy way" divides a decimal by a whole number. Same answer, far less work – and the exact form √n/n is what you should write in an exam.
Before calculators, every engineer's handbook listed √2, √3, √5… to many digits, but never 1/√2 – because 1/√2 = √2/2 could be found by one easy halving. Even today, keeping the root on top (√2/2 rather than 1/√2) avoids rounding errors when a result is used again and again, for example in electrical calculations where 1/√2 ≈ 0.707 appears constantly. And if a 2√3 m beam must be cut into 3 equal pieces, each piece is 2√3/3 m = 2/√3 m ≈ 1.155 m – you need division to say so.
5. Side box: identities work for irrational numbers too ഗുണനസമവാക്യം
In Class 8 you learned (x + y)(x − y) = x² − y². The textbook reminds you of the picture that proves it – and points out that the picture does not care whether x and y are fractions or irrational numbers.
Lengths may be irrational, but area is still length × breadth. So (x + y)(x − y) = x² − y² is true for irrational x and y as well – and so is every other identity you know. Example: (√2 + 1)(√2 − 1) = (√2)² − 1² = 2 − 1 = 1. You will use this in Exercise 2.
6. Check yourself
- Key idea 2: if √x × √y = √z then √z/√x = √y – e.g. √6/√2 = √3, √50/√2 = √25 = 5.
- Key idea 3: √(x/y) = √x/√y – e.g. √(3/4) = √3/2.
- Key idea 4: 1/√x = √x/x – rationalise the denominator: 1/√2 = √2/2 ≈ 0.707, 1/√3 = √3/3 ≈ 0.577.
- Trick for decimals: make the bottom a perfect square, √(1/2) = √(2/4) = √2/2.
- Identities such as (x + y)(x − y) = x² − y² are true for irrational numbers: (√2 + 1)(√2 − 1) = 1.
Next: Exercise 2 – all five questions solved with animations ▶