About Power Vs Exponent
Difference Between Power and Exponent
The main aspect that distinguishes power and exponents is that power implies the repeated multiplication of a factor or base, whereas an exponent denotes the number of times the factor or the base is multiplied to itself. A superscript is used to denote exponents. A superscript is a little digit placed above and to the right of a number in mathematics. For example, the power \(4^3\) is made up of 4 as the base and 3 as the exponent.
Extremely large or small numbers are difficult to express, compare and operate. Hence, the assistance of powers and exponents is used. For instance, exponents are used in scientific notations to represent huge or small quantities as powers of ten. For example, the distance between the Earth and the Sun is \(1.5~\times~10^8\) km when represented in terms of power.
When a number is:
- Squared – the exponent is 2
- Cubed – the exponent is 3
Quick Recall
Solved Examples
Example 1: Write the product of the given numbers as a power.
A) \(9~\times~9~\times~9~\times~9~\times~9~\times~9~\times~9\)
B) \(1000~\times~1000~\times~1000~\times~1000~\times~1000\)
C) \(17~\times~17~\times~17~\times~17~\times~17~\times~17\)
Solution:
A) \(9~\times~9~\times~9~\times~9~\times~9~\times~9~\times~9\)
Nine is multiplied seven times.
Hence, \(9~\times~9~\times~9~\times~9~\times~9~\times~9~\times~9~=~9^7\)
B) \(1000~\times~1000~\times~1000~\times~1000~\times~1000\)
1000 is multiplied five times.
Hence,
\(1000~\times~1000~\times~1000~\times~1000~\times~1000~=~1000^5\)
This can be further written as: \(1000^5~=~(10^3)^5~=~10^{15}\)
C)\(17~\times~17~\times~17~\times~17~\times~17~\times~17~=~17^6\)
17 is multiplied six times.
Hence, \(17~\times~17~\times~17~\times~17~\times~17~\times~17~=~17^6\)
Example 2: Verify if the given numbers are perfect squares.
A) 81
B) 31
C) 64
Solution:
A perfect square is defined as a number that can be expressed as a product of a number and itself.
A) \(9~\times~9~=~81\)
Nine times nine is 81. So, 81 is a perfect square.
B) 31 is not a perfect square, as it cannot be expressed as a product of a number by itself.
C) \(8~\times~8~=~64\)
Eight times eight is 64. So, 64 is a perfect square.
Example 3: In a sculptor’s workshop, the smallest sculpture is 3 cm tall. The height of each successive figure is three times the height of the previous sculpture. What are the different heights of the figures in the workshop if the height of the tallest figure is 243 cm?
Solution:
According to the question, the smallest figure is 3 cm and the largest figure has a height of 243 cm. Since height is progressing by multiplying the previous height by three, these are the heights of the sculptures:
1st sculpture: 3 cm
2nd sculpture: \(3^2~=~3~\times~3~=~9\) cm
3rd sculpture: \(3^3~=~3~\times~3~\times~3~=~27\) cm
4th sculpture: \(3^4~=~3~\times~3~\times~3~\times~3~=~81\) cm
5th sculpture: \(3^5~=~3~\times~3~\times~3~\times~3~\times~3=~243\) cm
So, the different heights of the sculptures in the workshop are 3cm, 9cm, 27cm, 81cm and 243cm.
Power is a mathematical term that refers to the value when a base number is multiplied by itself repeatedly and the exponent is the number of times the base number is multiplied.
The degree of a polynomial is the highest power of the variable in the polynomial but the exponent denotes how many times the base is multiplied to itself in a power.
A number when raised to 1 denotes the number itself. For example, \(3^1~=~3\).