Laws of Exponents for Real Numbers
Laws of Exponents are fundamental rules used in mathematics to simplify expressions involving exponents. These laws help in solving arithmetic problems efficiently by defining operations like multiplication, division, and more on exponents. In this article, we will discuss the laws of exponent for real number in detail with some solved examples.
Table of Content
- What are the Laws of Exponents?
- Summary
- Sample Problems
- FAQs
What are the Laws of Exponents?
Laws or rules which help us govern the calculation of exponents are called laws of exponent. Laws of Exponents for Real Numbers are:
- Product Law
- Quotient Law
- Power Law
- Power of Product Law
- Power of Quotient Law
- Zero Power Rule
- Exponent of Exponent
Let’s discuss the laws in detail as follows.
Product law
The Product law states that if the base of two numbers is same, their exponents can be added directly.
am × an = am+n
To verify this property, see how many times “a” is multiplied in total. It is multiplied “m + n” times. Thus, the property.
Example: a2a3 = (aa)(aaa) = a5
Quotient law
According to this law, if the two numbers in the numerator and denominator are the same, their exponents can be arranged in such a manner that the exponent in the denominator is subtracted by the exponent in the numerator
This can also be verified similar to the previous one, just see the number of times a is multiplied and then reduce that number by the number of times it is divided.
Example:
Power Law
According to Power-law, if an exponent is a power of another exponent, we can simply multiply the exponents.
(am)n = amn
First, multiply “a” m times, and then do this operation n times.
Example: (x3)2 = (xxx)2 = (xxx)(xxx) = x6 = x(3 × 2)
Power of Product law
According to the power of product law, if two real numbers say, a and b here are multiplied and raised to power m, we can distribute the exponent to both a and b separately.
am x bm = (ab)m
This property is just a rearrangement of all these variables.
Example: a3 × b3 = (aaa) (bbb) = (ab)(ab)(ab) = (ab)3
Power of Quotient law
According to the power of quotient law, if two real numbers are in numerator and denominator and are raised to power n, the power can be separately distributed to both the numbers.
This also can be verified by a simple rearrangement of the variables.
Example:
Zero Power Rule
As long as x is not equal to zero, raising it to the power of zero should give us 1 as result.
a0 = 1, a≠ 1.
Let’s take an example to make this more evident,
1 =
Note:
Expression 00 is considered to be indeterminate. Why? Because there are two answers to it.
We know, x0 = 1, So, 00 = 1.
We also know, 0x = 0. So, 00 = 0.
These are two contradictory answers, thus we consider 00 to be indeterminate.
Exponent of Exponent
Sometimes in more complex scenarios, exponents over exponents are given. Let’s see how to approach them,
We will start solving from up,
Let’s look at some examples and questions where these properties are applicable.
Summary: Laws of Exponents for Real Numbers
We will now see the all laws of exponents in a tabular form.
Law | Formula |
---|---|
Product law |
|
Quotient law |
|
Power Law | |
Power of Product law | |
Power of Quotient law | |
Zero Power Rule |
, |
Negative Exponent Rule |
Read More,
Sample Problems: Laws of Exponents
Problem 1: Find the value of 2-3
Solution:
Problem 2:
Solution:
Problem 3: Simplify using above properties:
(-4)5 x (-4)-10
Solution:
(-4)5 x (-4)-10
= -4(5-10) (am × an = am+n)
= (-4)-5
Problem 4: Simplify using above properties:
Solution:
Problem 5: Simplify,
Solution:
Problem 6: What is the 5th root of 515.
Solution:
We know that nth of a number “a” is represented by
Thus, 5th of 515 will be given by,
= 53
= 125
Problem 7: Find the value of n
Solution:
⇒ 2n-3 × 24n – 16 = 1
⇒ 2n – 3 + 4n – 16 = 20
Comparing Both Sides
n – 3 + 4n -16 = 0
⇒ 5n -19 = 0
⇒ n =
FAQs: Laws of Exponents for Real Numbers
What are the Laws of Exponents?
The Laws of Exponents are a set of rules that describe how to manipulate expressions involving exponents.
What is the Law of Product of Powers?
The Law of Product of Powers states that when multiplying two powers with the same base, you add their exponents. For example, am × an = am+n.
What is the Law of Power of a Power?
The Law of Power of a Power states that when raising a power to another exponent, you multiply the exponents. For example, (am)n = amn.
What is the Law of Power of a Product?
The Law of Power of a Product states that when raising a product to a power, you can distribute the exponent to each term inside the parentheses. For example, (ab)n=an × bn.
What is the Law of Quotient of Powers?
The Law of Quotient of Powers states that when dividing two powers with the same base, you subtract their exponents. For example, an/am = am−n.
What is the Law of Zero Exponent?
The Law of Zero Exponent states that any nonzero base raised to the power of zero is equal to 1. For example, a0 = 1.
What is the Law of Negative Exponent?
The Law of Negative Exponent states that any nonzero base raised to a negative exponent is equal to the reciprocal of the base raised to the positive exponent. For example, a−n = 1/an.
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