Exponents
Nov 30, 2022, 16:45 IST
An introduction to the exponents explains the representation of vast or very small numbers. An exponents are a value that is raised to a number. Both the exponents and powers are related to each other. 7,350,000,000,000,000,000,000,000 kg is the mass of the moon. But the value of the moon is so huge that we cannot read it easily. Exponents were introduced to make it easy to read and write such huge numbers.
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What are Exponents?
An expression consisting of a repeated multiplication power of the same factor is known as an exponent. It is also called raised to the power of a number or indices. Consider an example like 42; the number 4 is called the base, whereas 2 is the exponent of the expression. The value of the expression is derived by multiplying the base as many times as the number of power. In the example above, the power is 2. Thus the value becomes 4 x 4 = 16.
Types of Exponent
Exponents can be divided into four types that are based on the number in power. They are
- Positive Exponents
Positive exponents can be simplified by just multiplying the base by itself the number of times indicated by the exponent/power.
- Negative Exponents
A negative exponent can be simplified by just placing 1 in the numerator and the base along with the exponent in the fraction's denominator.
- Zero Exponents
Any expression with the exponent as 0 is equal to 1, and no need to consider the base value during simplification.
- Rational Exponents
Rational or fractional exponents will become both radical or roots. For example, 41/3 can be written as 3 root of 4, and 85/2 can be written as 2 root (or square root) of 8 raises to the power 5.
What are Negative Powers?
The negative power is similar to the positive power of the exponent. The normal change in the negative power is that the value of the expression is the reciprocal of the value that is obtained in the positive case.
The negative power can be written as-
6-2
1/62 = 1/36
Relation Between Positive and the Negative Power
Let's consider a base number to be x and the power to be a,
The relation between both the positive and the negative expression is given as:
Xa = 1/x-a
Value of expression when the value of base is 0 or 1:
Both 0 and 1 are the only power that gives the exact same value, i.e. 0 and 1 respectively, for the different exponent values.
Such as
01234567 = 0
1123454 = 1
Laws of Exponents
| Product law | am.an = am+n |
| Quotient law | am/an = am-n |
| Power raised to a power | (am)n = amn |
| Product to a power | an.bn = (ab)n |
| Quotient to a power | an/bn = (a/b)n |
| Zero power | a0 = 1 |
| Negative exponent | a-m = 1/am |
| Fraction exponent | a1/n = n√a |
Solved Examples
Q1. What is the exponential form of 256? Mention its base and exponent.
Ans. 256 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2
= 28
Thus, the base is “2” and the exponent is “8”.
Q2. Which one is smaller 35 or 53
Ans. 35 = 3 x 3 x 3 x 3 x 3
= 243
53 = 5 x 5 x 5
= 125
Therefore, 35 > 53
Frequently Asked Question (FAQs)
Q1. What are the properties of exponents?
Ans. The Five Exponent Properties-
- Product of Powers
- Power to a Power
- Quotient of Powers
- Power of a Product
- Power of a Quotient
Q2. How many exponent rules are there?
Ans. Seven exponent rules
Q3. What is the value of exponents?
Ans. The value of the exponent is totally based on the number of times the base is multiplied by itself.
Q4. Can an exponent be negative?
Ans. A positive exponent tells how many times to multiply a base number, whereas a negative exponent tells how many times to divide a base number.
Q5. Can you divide an exponent?
Ans. To divide them, you take the exponent value in the numerator (the top exponent) and subtract the exponent value of the denominator (the bottom exponent).
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