Experts have designed these Class 8 Maths Notes and Chapter 6 We Distribute, Yet Things Multiply Class 8 Notes for effective learning.
Class 8 Maths Chapter 6 Notes We Distribute, Yet Things Multiply
Class 8 Maths Notes Chapter 6 – Class 8 We Distribute, Yet Things Multiply Notes
We use multiplication in our daily life, whether we are counting money, planning parties, or solving puzzles. But sometimes, numbers are big or tricky. That’s where the distributive property helps! It lets us break a big problem into smaller parts to make multiplying easier and smarter. By using this property, we multiply numbers by separating them into parts, rearranging them, and solving them in quicker way.
→ Algebraic identities help us solve big calculations quickly using patterns and formulas.
→ The general form of distributive property is (a+m) × (b+n) = ab + mb + an + mn
→ The identity (a+b)2 = a2 + 2ab + b2 is used to calculate total area when two parts are combined.
→ The identity (a-b)2 = a2 – 2ab + b2 helps when subtracting smaller values from bigger ones-like shrinking squares or cutting lengths.
→ The identity (a+b) ×(a-b)=a2-b2 is used to find the difference between two square areas or values quickly.
→ Identities are useful in mental math tricks, like finding squares of numbers, multiply the numbers. Architects and designers use them to plan tiles, floor spaces, or wall patterns without measuring every time.
→ The distributive property, a(b+c) = ab + ac, is used to split quantities while shopping, cooking, or budgeting.
→ Distributive property also helps simplify expressions in algebra and break large multiplications into easier parts.
In earlier classes, we have studied about algebraic expressions. A combination of constants and variables connected by four fundamental arithmetic operators +, x and + is called an algebraic expression.
e.g. 2x – 3, 2xy + 5 etc.
Here, we will learn about distributive property and some important identities of algebraic expression as
- (a + b)2 = a2 + 2ab + b2
- (a – b)2 = a2 – 2ab + b2
- a2 – b2 = (a + b) (a – b)
Distributive Property
The distributive property is an algebraic property that is used to multiply a single value and two or more values within a set of brackets.
The distributive property states that when a factor is multiplied by sum or addition of two terms, it is essential to multiply each of the two numbers by factor and finally perform the addition operation.
Distributivity of Multiplication over Addition
If a, b and c are three integers then
a × (b + c) = (a × b) + (a × c)
e.g. (-2) × (3 + 5) = [(-2) × 3] + [(-2) × 5]
= (-6) + (-10)
= -16
Distributivity of Multiplication over Subtraction
For any three integers a, b and c, we can say a × (b – c) = (a × b) – (a × c)
We usually skip writting the ‘x’ symbol before or after brackets, just as in the care of expressions like 3a, pq etc.
Standard Expansions Using Distributivity
For any three integers a, b and c,
(a + b)c =ac + bc
For any four integers a, b, c and d,
(a +b)(c + d) = (a + b) c + (a + b)d
= ac + bc + ad + bd
Some Rules for Products
Let the product of twd numbers a and b be ab.
(i) If one of the numbers say b, is increased by 1 then
a(b + 1 ) = ab + a
Thus, the product ab increases by a, when b is increased by 1.
Similarly, if one of the numbers say a, is increased by 1 then
(a + 1) b = ab + b
Thus, the product ab increases by b, when a is increased by 1.
(ii) If both numbers a and b are increased by 1 then
(a + 1) (b + 1) = (a + 1) b + (a + 1) 1 [by distributive property]
= ab + b + a + 1
= ab + (b + a + 1)
Thus, the product ab increases by a + b + 1, when each of a and b are increased by 1.
(iii) If a is increased by m and b is increased by n then
(a + m) (b + n) =(a + m) b + (a + m)n
= ab + mb + an + mn
= ab + (mb + mn + an)
Thus, the product ab increases by (mb + mn + an), when a is increased by m and b is increased by n.
(iv) If a is increased by 1 and b is decreased by 1 then
(a + 1) (b – 1) = (a + 1) b – (a +1) 1
= ab + b – a – 1
= ab + (b – a – 1)
This shows how the original product ab is affected by the increase in a and decrease in b.
The net change in the product is given by b – a -1. So, depending on the value of b – a – 1, the final product may increase or decrease.
(v) If a is increased by m and b is decreased by n then
(a + m)(b – n) = (a + m)b – (a + m)n
= ab + mb – an – mn
= ab + (mb – an – mn)
Thus, the product ab is affected by mb – an – mn when a is increased by m and b is decreased by n.
The net change in the product is given by mb – an – mn.
So, depending on the values of m, n, a and b, the final product may either increase or decrease or remains same.
Special Cases of the Distributive Property
Square of Sum of Two Numbers
The square of the sum of two numbers is equal to the sum of the squares of those two numbers plus twice their product.
Let the two numbers be a and b.
Then, square of sum of a and b = (a + b)2
= (a + b)(a + b)
= (a + b) a + (a + b)b
[by distributive property]
= a2 + ba + ab + b2
= a2 + ab + ab + b2 [∵ ab = ba]
= a2 + 2 ab + b2
Square of Difference of Two Numbers
The square of the difference of two numbers is equal to the square of the first number plus the square of the second number minus twice the product of the two numbers.
Let the two numbers be a and b.
Then, square of difference of a and b
= (a – b)2 = (a – b)(a – b)
= (a – b) a – (a – b)b
[by distributive property]
= a2 – ba – ab + b2
= a2 – ab – ab + b2[∵ab = ba]
= a2 – 2ab + b2
Sum of Squares Relationship
Let the two numbers be a and b.
Then, (a + b)2 = a2 +2ab + b2 …(i)
and (a – b)2 = a2 – 2ab + b2 … (ii)
On adding Eqs (i) and (ii), we get
(a + b)2 + (a – b)2 – a2 + 2ab + b2 + a2 – 2ab + b2
⇒ (a + b)2 + (a – b)2 = 2a2 + 2b2
⇒(a + b)2 + (a – b)2 = 2(a2 + b2)
Now, on subtracting Eq. (ii) from Eq. (i), we get
(a + b)2 – (a – b)2 = a2 + 2ab + b2 – a2 + 2ab – b2
⇒ (a + b)2 – (a – b)2 = 4ab
Product of Sum and Difference
The product of the sum and difference of two numbers is equal to difference of their squares.
Let the two numbers be a and b.
Then, product of sum and difference = (a + b)(a – b)
= (a + b)a – (a + b)b
[by distributing property]
= a2 + ba – ab – b2
= a2 + ab – ab – b2 [∵ ab = ba]
= a2 – b2
Rule for multiplying by 11,101, 1001, 10001,…
If a be any number multiply by 11, 101, 1001, 10001, …………..
then the result will be a × 10k+1 + a, where k is number of zeros between two Is.
Multiply by 99
If a be any number then
a × 99 = a × (100 – 1) = a × 100 – a
Multiply by 999
If a be any number then
a × 999 = a × (1000 – 1) – a × 1000 – a