Exponents

Powers and Exponents

The power of a number indicates the number of times it must be multiplied. It is written in the form a^b . Where ‘ b ’ indicates the number of times ‘ a ’ needs to be multiplied to get our result. Here ‘ a ’ is called the base and ‘ b ’ is called the exponent .

For example: Consider 9³. Here the exponent ‘3’ indicates that base ‘9’ needs to be multiplied three times to get our equivalent answer which is 729.

Powers with Negative Exponents

A negative exponent in power for any non-integer is basically a reciprocal of the power.

Explanation:
For a non-zero integer a with exponent -b :
a^-b = 1 a^b

Comprehending Powers and Exponents

Consider the following table:

Expanding a Rational Number Using Powers

A given rational number can be expressed in expanded form with the help of exponents. Consider a number 1204.65. When expanded the number can be written like:
1204.65 = 1000 + 200 + 4 + 0.6
1204.65 = + 0.05
1204.65 = (1 × 10³) + (2 × 10²) + (0 × 10¹) + (4 × 10⁰) + (6 × 10^-1) + (5 × 10^-2) 1204.65 = (1 × 10³) + (2 × 10²) +
1204.65 = (0 × 10¹) + (4 × 10⁰) +
1204.65 = (6 × 10^-1) + (5 × 10^-2)

Laws of Exponents

Exponents with like Bases

Given a non-zero integer a:
a^m × aⁿ = a^m+n where m and n are integers.
a^maⁿ = a^m-n where m and n are integers.
Examples:
2³ × 2⁷ = 2¹⁰
2⁷2³ = 2⁴

Power of a Power

Given a non-zero integer a, (a^m )ⁿ = a^mn , where m and n are integers.
For example: (2⁴ )³ = 2^4×3 = 2¹²

To the power of zero

Given a non-zero integer a, a⁰ = 1
Any number to the power 0 is always 1.
Note: 0⁰ simplifies to 0 0 as 0⁰
Note: 0⁰ simplifies to 0 0 a= 0^1-1
Note: 0⁰ simplifies to 0 0 a= 0¹ 0¹
Note: 0⁰ simplifies to 0 0 a= 0 0
which we know is undefined and therefore 0⁰ is also undefined .

Exponents with Unlike Bases and Same Exponent

Given non-zero a,b:
a^m × b^m = (a × b)^m
Example: 2³ × 5³ = 10³ = 1000

Uses of Exponents

Inter Conversion between Standard and Normal Forms

Very large numbers or very small numbers can be represented in the standard form with the help of exponents .

If it is a very large number like 150,000,000,000 then we need to count the total digits after the first digit . And when we do so the exponent will be that total number .
In, 150,000,000,000 there are 11(2+3+3+3) digits easily counted with the help of periods.
So the standard form would be
1.5×10¹¹ OR
1.5e+11(Easier to type on keyboard)

Notation Breakdown:
I- Standard Form: 1.5 × 10¹¹
II- Digital Form: 1.5e+11

Key Components:
• e = ×10^ (shorthand for base-10 exponent)
• +/- = Sign of exponent (e+11 = ×10¹¹ , e-6 = ×10^-6 )
• Identical value: 1.5 × 10¹¹ = 1.5e+11

Usage Context:
- Standard Form: Used in textbooks, research papers, and formal writing
- Digital Form: Used in programming (Python/Excel), calculators, and data files

If it is a very small number like 0.000007, we need to count the total digits before the last digit in order to represent the number in its standard form. The exponent will be negative of that total number .

In this case,the exponent will be negative 6 as we counted 6 total digits before the last digit.
Therefore our standard form representation will be
7×10⁻⁶ OR
7e-6

The exponents are also useful when converting the number from it’s standard form to it’s natural form.

Comparison of Quantities Using Exponents

In order to compare two large or small quantities , we convert them to their standard exponential form and divide them.

For example : To compare the diameter of the earth and that of the sun.
Diameter of the Earth = 1.2756 × 10⁶ m
Diameter of the Sun = 1.4 × 10⁹ m
So quotient for comparision
= 1.4 × 10⁹ m 1.2756 × 10⁶ m
= 1.4 × 10⁹ 1.2756 × 10⁶ (On canceling the unit meter)
= 1.4 × 10⁹  ÷ 10⁶ 1.2756
= 1.4 × 10^(9-6) 1.2756
= 1.4 × 10³ 1.2756
= 1.4 × 1000 1.2756
= 1400 1.2756
≈ 1097.5
So the diameter of the Sun is 1097.5 times that of the Earth! While calculating the total or the difference between two quantities, we must first ensure that the exponents of both quantities are the same.

MCQs

1. What is the value of 2³?

(a) 6

(b) 8

(c) 9

(d) 4

► (b) 8

2. 5⁰ equals:

(a) 0

(b) 1

(c) 5

(d) Undefined

► (b) 1

3. What is 3² × 3³?

(a) 3⁵

(b) 3⁶

(c) 3⁷

(d) 3¹⁵

► (a) 3⁵

4. (2⁴)² equals:

(a) 2⁶

(b) 2⁸

(c) 2⁴

(d) 2¹²

► (b) 2⁸

5. 7⁻¹ is equal to:

(a) 7

(b) −7

(c) 17

(d) 0

► (c) 17

6. aᵐ × aⁿ =

(a) aᵐ⁺ⁿ

(b) aᵐ⁻ⁿ

(c) aᵐⁿ

(d) aᵐ/ⁿ

► (a) aᵐ⁺ⁿ

7. The value of (−2)³ is:

(a) 8

(b) −8

(c) 6

(d) −6

► (b) −8

8. x⁰ =

(a) 1

(b) 0

(c) x

(d) Undefined

► (a) 1

9. (3²)³ =

(a) 3⁵

(b) 3⁶

(c) 6⁵

(d) 9³

► (b) 3⁶

10. What is the reciprocal of 10⁻²?

(a) 10²

(b) −10²

(c) 110²

(d) 1100

► (a) 10²

11. Simplify: 2⁵2³

(a) 2²

(b) 2⁸

(c) 2³

(d) 2⁰

► (a) 2²

12. Which of the following is in exponential form?

(a) 5 × 5 × 5

(b) 3³

(c) 2 + 2

(d) 7 × 2

► (b) 3³

13. The standard form of 5000000 is:

(a) 5 × 10⁵

(b) 5 × 10⁶

(c) 5 × 10⁷

(d) 50 × 10⁶

► (b) 5 × 10⁶

14. 4⁻² =

(a) −8

(b) 116

(c) 16

(d) −14

► (b) 116

15. aⁿaᵐ =

(a) aⁿ⁺ᵐ

(b) aⁿ⁻ᵐ

(c) aᵐ⁻ⁿ

(d) aⁿ × aᵐ

► (b) aⁿ⁻ᵐ

16. What is the value of 10⁻³?

(a) 1000

(b) −1000

(c) 11000

(d) 0.001

► (c) 11000

17. Simplify: (5³ × 5⁴)

(a) 5⁷

(b) 5¹²

(c) 5¹

(d) 5⁶

► (a) 5⁷

18. 2⁻⁵ is equal to:

(a) 132

(b) −32

(c) 32

(d) 125

► (a) 132