Power Play Class 8 Notes Maths Chapter 2

Experts have designed these Class 8 Maths Notes and Chapter 2 Power Play Class 8 Notes for effective learning.

Class 8 Maths Chapter 2 Notes Power Play

Class 8 Maths Notes Chapter 2 – Class 8 Power Play Notes

Writing very large and small numbers, such as the number of atoms in a speck of dust or the distance to the nearest star in miles, is not easy. We would end up with a string of zeros so long it would be tedious to write, hard to read, and almost impossible to calculate. This is where the “power play” of mathematics comes in – the elegant and efficient concept of exponents and powers.

Understanding exponents is a fundamental step in mastering advanced mathematics. It is a tool that enables us to handle numbers more effectively and in a more sophisticated manner, unlocking the secrets of growth, decay, scientific notation, and much more.

→ An exponent (also called an index or power) is a shorthand way to represent repeated multiplication of the same number. Instead of writing 2 × 2 × 2 × 2 × 2, we can simply write 2⁵.

→ Base: The number that is being multiplied repeatedly. In 2⁵, the base is 2.

→ Exponent (or Power/Index): The small number written above and to the right of the base. It tells us how many times the base is used as a factor in the multiplication. In 2⁵, the exponent is 5.

→ Power: The entire expression, 2⁵, is called a power. We read 2⁵ as ” 2 to the power of 5 ” or ” 2 to the fifth power,”

→ In general, n^a is n × n × n × n × n × …… × n ( n multiplied by itself a times) and n^-a = \(\frac{1}{n^a}\)

→ Laws of Exponents

• n^a × n^b = n^a+b
• (n^a)^b = (n^a)^b = n^a×b
• n^a ÷ n^b = n^a-b (n ≠ 0)
• n^a × m ^a = (n × m)^a
• n^a ÷ m^a = (n ÷ m)^a (m ≠ 0)
• n⁰ = 1 (n ≠ 0)

→ The scientific notation for the number 308100000 is 3.081 × 10⁸. The standard form of the scientific notation of any number is x × 10y, where x ≥ 1 and x < 10, and y is an integer.

In earlier classes, we learned about multiplication and repeated multiplication of numbers. Building on that, this chapter introduces powers and exponents, where repeated multiplication is written in a compact form.

Understanding Exponential Growth

Exponential growth occurs when a quantity multiplies by the same factor repeatedly.
e.g. Folding a paper of thickness 0.001 cm

• After 1 fold : 0.001 × 2 = 0.002 cm
• After 2 folds : 0.001 × 2 × 2 = 0.004 cm
• After 3 folds : 0.00 1 × 2 × 2 × 2 = 0.008 cm
  ……………………………………………………..
  ……………………………………………………..
• After 10 folds : 0.001 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
  = 1.024 cm

Exponential Notation and Operations
Repeated multiplication is written using powers.
a × a × a… (n times) = aⁿ
where, a = base and n = exponent (or power)
e.g. 5⁴ = 5 × 5 × 5 × 5 = 625
(-2)³ = (-2) × (-2) × (-2) = – 8

Laws of Exponents
If a and b are non-zero integers and their exponents m and n are also integers then
(i) a^m × aⁿ = a^m+n
e.g. 2² × 2² = 2⁽²⁺³⁾ = 2⁵

(ii) a^m ÷ aⁿ = a^m-n
e.g. 5³ ÷ 5² = 5^(3-2) = 5¹ = 5

(iii) (a^m)ⁿ = a^mn or (aⁿ)^m
e.g. (3²)⁴ = 3⁸

(iv) a^m × b^m =(ab)^m
e.g.2⁴ × 4⁴ = (2 × 4)⁴ = 8

(v) \(\frac{a^n}{b^n}=\left(\frac{a}{b}\right)^n\)
e.g. \(\frac{7^3}{5^3}=\left(\frac{7}{5}\right)^3\)

(vi) a^-n = \(\frac{1}{a^2}\)
e.g. (5^-2) = \(\frac{1}{5^2}\)

(vii) a° = 1 e.g. 8° = 1
Note In general, (-1)ⁿ = 1 if n is an even integer and (-1)ⁿ = -1 if n is an odd integer.

Exponential form of Large Numbers
Large numbers can be written as power of prime factors.

Powers of 10
The ‘Powers of 10’ refer to expressing numbers as multiple of 10 raised to various exponents, which indicates the place value of each digit in an expanded form.
e.g. 38561 = (3 × 10000) + (8 × 1000) + (5 × 100) + (6 × 10) + 1
= (3 × 10⁴) + (8 × 10³) + (5 × 10²) + (6 × 10¹) + (1 × 10°)
[∵ a° = 1]

Scientific Notation or Standard Form
In scientific notation, a number is expressed as the product of number between 1 and 10 and a power of 10.
The standard form or the scientific notation of any number is n x 10*, where 1 < n < 10 and x is an integer.
This method is used to represent very large or very small numbers in a more concise form.

Comparing very large and very small numbers
We can compare very large and very small numbers very easily. This is illustrated with the help of example given below.