In view of the coronavirus pandemic, we are making LIVE CLASSES and VIDEO CLASSES completely FREE to prevent interruption in studies
Maths > Number Systems > Laws of Exponents for Real Numbers
Number Systems

Laws of Exponents for Real Numbers

Supposed you are asked to calculate or multiply large numbers and you don’t have an electronic calculator. What if you get the wrong answer? How can we make this calculation easier? So let us try to understand some of the exponent rules for real numbers.

Suggested Videos

Play
Play
Play
Arrow
Arrow
ArrowArrow
Properties of irrational numbers
Rational numbers between two rational numbers
Introduction to decimal numbers
Slider

 

Laws of Exponents or Exponent Rules

What is meant by an exponent? You must have come across the expression  3². Here 3 is the base and 2 is the exponent. Exponents are also called Powers or Indices. The exponent of a number tells how many times to use the number in a multiplication. Let us study the laws of exponent. It is very important to understand how the laws of exponents laws are formulated.

Exponent Rules

(Source: math warehouse)

1. Product law

According to the product law of exponents when multiplying two numbers that have the same base then we can add the exponents

a× an= a m+n

where a, m and n all are natural numbers. Here the base should be the same in both the quantities. For example,

  • 2³ × 2= 27
  • 22/3 × 21/5 = 2 2/3 + 1/5   = 2(10+3)/15 . We get, = 212/15
  • (-6) 3 x (-6) = (-6) 3+2 = (-6) 5

Browse more Topics under Number Systems

2. Quotient Law

According to the quotient law of exponents, we can divide two numbers with the same base by subtracting the exponents. In order to divide two exponents that have the same base, subtract the power in the denominator from the power in the numerator.

a÷ a= a m-n

where a, m and n all are natural numbers. Here the base should be the same in both the quantities. For example,

  • 25 ÷ 2= 2²
  • p6 ÷ p= p 6 – 2  = p 4

3. Power Law

According to the power law of exponents if a number raise a power to a power, just multiply the exponents

(am)= am×n

Here there is one base a  and two powers m and n. For example, ( 53 )= 53×2 = 56

Important Points to Remember on Exponent Rules

  • \( \frac{1}{a^n} \) = a-n. A non zero base raised to a negative exponent is equal to the reciprocal of the base raised to the positive exponent or \( \frac{1}{a^-n} \) = a+n
  • a= 1. This says that anything raised to the zero power is 1. For example, 50= 1, (1000) 0 = 1
  • a= a

Power of Product

The power of product rule states that: (ab)m  = a× bm, a and b are positive real numbers and m is the rational number. For example, ( 2 × 5)10 = 210 × 510

Power of Quotient

The power of product rule states that:
\( \frac{a}{b} \)^n = \( \frac{a^n}{b^n} \)
Or, \( \frac{2}{5} \)^12 = \( \frac{2^12}{5^12} \)

Solved Questions on Exponent Rules

Question 1: If 12= 3× 4calculate the value of t.

  1. 2.5
  2. 5
  3. 10
  4. 20
  5. 40

Answer : The correct option is “B”. Given: 12= 3× 4t. If two terms have the same power and they are multiplied then the power can be taken as common = 12=( 3 × 4 )= 12= 12t. If two terms have the same base, then we can equate their powers.

Question 2: Which of the following expresses the power law of exponents?

  1. a× an= a m+n
  2. a× bn= (ab) m
  3. a= 1
  4. (am )n= a mn

Answer : The correct answer is “D”. The power law states that to raise a power to a  power, just multiply the exponents. Therefore, (am)n= a m×n = amn. So, D is the correct option.

Question 3: What is difference between power and exponent?

Answer: We often call exponents powers or indices. Power refers to an expression which is representing repeated multiplications of the same number. While exponent is the quantity which is representing the power to which we raise the number. We use both these terms interchangeably in math.

Question 4: What is the zero exponent rule?

Answer: When there is a number or variable that is raise to a power, the number (or variable) is referred to as the base whereas the superscript number refers to the exponent or power. Thus, the zero exponent rule states that any base having an exponent of zero equals to one.

Question 5: What are the three laws of exponent?

Answer: The three laws of exponent are firstly, multiplication of identical bases and addition of exponents. Secondly, dividing the identical bases and subtracting the exponent. Thirdly, multiplication of exponent when two or more exponents and just one base is present.

Question 6: What is exponential form?

Answer: Exponential form refers to a numeric form that involves exponents. One method to write such a number is by identifying that each position is representing a power (exponent) of 10. Thus, you can initially break it up into different pieces

Share with friends

Customize your course in 30 seconds

Which class are you in?
5th
6th
7th
8th
9th
10th
11th
12th
Get ready for all-new Live Classes!
Now learn Live with India's best teachers. Join courses with the best schedule and enjoy fun and interactive classes.
tutor
tutor
Ashhar Firdausi
IIT Roorkee
Biology
tutor
tutor
Dr. Nazma Shaik
VTU
Chemistry
tutor
tutor
Gaurav Tiwari
APJAKTU
Physics
Get Started

2
Leave a Reply

avatar
2 Comment threads
0 Thread replies
0 Followers
 
Most reacted comment
Hottest comment thread
2 Comment authors
Olak JamohAmulya Recent comment authors
  Subscribe  
newest oldest most voted
Notify of
Amulya
Guest
Amulya

Satisfied with the answer

Olak Jamoh
Guest
Olak Jamoh

Very clear. ..thanks

Stuck with a

Question Mark?

Have a doubt at 3 am? Our experts are available 24x7. Connect with a tutor instantly and get your concepts cleared in less than 3 steps.
toppr Code

chance to win a

study tour
to ISRO

Download the App

Watch lectures, practise questions and take tests on the go.

Get Question Papers of Last 10 Years

Which class are you in?
No thanks.