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Laws of Exponents

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Exponents are a way of representing very large or very small numbers. Exponent rules are the laws of the exponents that are used to solve various exponents’ problems. The multiplication, division, and other operations on exponents can be achieved using these laws of exponents. There are different rules of exponents also called laws of exponents in Mathematics and all these laws are added in the article below.

In this article, we will learn about Exponents Definition, Laws of Exponents, Laws of Exponents Examples, and others in detail.

Exponents Definition

When a number is raised to some power then the power on the base number is called Exponent. Exponent simply means a base number is multiplied by itself equal to the power mentioned on it.

For Example, if we say Pn this means P is multiplied by itself ‘n’ a number of times. It can be expanded as P×P×P×P×P×P . . . n times.

For example, 53 = 5 × 5 × 5 = 125; the equation is read as “five to the power of three.” 

If the exponent is 2 then it is also known as “squared,” whereas if the exponent is 3 it is known as “cubed.” When calculating the area, the term ‘squared’ is used because we multiply length (m/cm) two times and in the case of volume the term ‘cubed’ is used as we multiply length (unit = m/cm) three times.

Exponent helps us to write very large as well as very small quantities. For instance, we can write large quantities such as the Mass of the Earth which is 5.97219×1024 kg as well as very small quantities such as the Mass of the Electron which is 9.1×10-31 kg.

What are Exponent Rules?

Exponent rules are the rules that are used to solve exponent’s problems. Suppose we are given two exponents am and an and we have to find the product of the two exponents then we use the concept of exponents rule or product of expoents rule, i.e.

am × an = a(m+n)

There are various other rules that are used to solve exponents problems. These rules are called the exponets rule.

What are Laws of Exponents?

Laws of Exponents are the set of rules that help us to solve arithmetic problems in an easy manner. Since at times, we can get large exponents that make multiplication lengthy then with the help of laws of exponents, we can solve the problems easily and in time bound manner.

Following are the seven Laws of Exponents that we must know to solve arithmetic problems involving exponents:

  • Product of Powers Rule
  • Quotient of Powers Rule
  • Power of a Powers Rule
  • Power of a Powers Rule
  • Power of a Quotient Rule
  • Zero Power Rule
  • Negative Exponent Rule

Product of Powers Rule

In the Product of Powers Rule, if two numbers with the same bases and different exponents are multiplied then exponents of the base are added to find the product. It is represented as xm×xn = x(m+n)

Example: 52 × 53 =?

Keep the base values the same because they’re both five, and then add the exponents together (2+3).

52 × 53 = 52+3 = 55

To get the answer, multiply five by itself five times.

55 = 5 × 5 × 5 × 5 × 5 = 3125

Quotient of Powers Rule

In Quotient of Powers Rule, if two numbers with the same bases and different exponents are divided then the exponents of the base are subtracted to find the quotient. It is represented as xa÷xb = x(a-b)

Example: 45 ÷ 43 =?

Solution:

45 ÷ 43 =?

Because both bases in this equation are four, they remain the same. Then subtract the divisor from the dividend using the exponents.

45 ÷ 43 = 45-3 = 42

Finally, if necessary, simplify the equation.

42 = 4 × 4 = 16

Power of a Power Rule 

In Power of a Power Rule, if a number raised to some power is again raised to some power then the two powers will be multiplied. It is represented as (xm)n = xm×n

Example: (23)2=?

Solution:

(23)2=?

Multiply the exponents together in equations like the one above while keeping the base constant.

23×2 = 26

However, we have to keep in mind that ((2^3)^2 ~\neq~2^{3^2} as (23)2 = 26 but 2^{3^2} = 2^9 as only exponent 3 is again raised to exponent 2 and not the whole number including base. 

Power of a Product Rule

In Power of a Product Rule, two different bases are raised to the same power are multiplied, then, bases are multiplied and power is common to the product of the bases. It is represented as (xm × ym) = (xy)m. If the given question is (xy)m then distribute the exponent to each portion of the base when multiplying any base by an exponent, hence (xy)m = (xm × ym)

Example: 23×33 =?

Solution:

Since the bases are different and the power is same then multiply the bases and raise it to the common power.

Therefore, 23×33 =(2×3)3 = 63 = 216

Example: (2×3)3 =? 

Solution:

In this case separate the same power to individual bases.

Hence, (2×3)3 = 23×33 = 8×27 = 216

Power of a Quotient Rule

In Power of a Quotient Rule, if two different bases with the same power are divided then the result is the quotient of the bases raised to the same power. This is represented as xm/ym= (x/y)m. In this case, vice versa is also true i.e. if the both numerator and denominator are raised to the same power then power is distributed to both numerator and denominator individually. It can be represented as (x/y)m =  xm/ym

Example: Simplify 64/34.

Solution:

In this case, find the quotient of the bases and raise common power to it.

64/34 = (6/3)4 = 24 = 16

Example: Simplify (6/3)4.

Solution:

In this case, distribute the power 4 to both the numerator and denominator.

(6/3)4 = 64/34 = (6×6×6×6)/(3×3×3×3) = 2×2×2×2 = 16

Zero Power Rule

In Zero Power Rule, if any base is raised to power zero, then the result will be 1. This can be represented as x0 = 1. Zero Power rule can be understood from the following description

Suppose we have to prove x0 = 1.

x0 = xn-n , where (0 = n-n)

From the Quotient of Power Rule, we know that if the base are same then we subtract the exponents while finding the quotient; the vice versa of Quotient of Power Rule also holds true. 

⇒ xn-n = xn/xn = 1

Hence, x0 = 1. 

Let’s consider an example for better understanding of the law.

Example: (1001)0 =?

As per Zero Power Rule, any number raised to power zero results the value 1.

(1001)0 = 1

Negative Exponent Rule

In Negative Exponent Rule, if a number is raised to negative interest then we convert the base to its reciprocal, and the power is changed to positive. The vice versa is also true i.e. if the exponent is positive and if the base is converted to its reciprocal then the exponent is changed to the negative value. It can be represented as (x/y)-m = (y/x)m

Example: (2/3)-2 =?

Solution:

Since, the exponent is negative the base is converted to its reciprocal.

⇒ (2/3)-2 = (3/2)2 = 32/22 = 9/4

Fractional Exponent Rule (Laws of Exponents with Fractions)

Fractional Exponent rule is a rule that is used to solve fractional exponents or the exponents that are in fractional form. An exponent in fractional form is written as a1/n and is read as nth root of a. It is also represented as,

a1/n = n√(a)

Here, a is the base of exponent and 1/n is the exponent in fractional form.

For example, simplify (8)1/3

= (8)1/3 = ∛(8)

= ∛(2×2×2)

= 2

Other Rules of Exponents

Apart from the above Seven Rules of Exponents, the following are some other Rules of Law of Exponents that we need to keep in mind while solving the questions of exponents.

  • If a negative number is raised to even number power then the result will be positive and if a negative number is raised to odd number power then the result is always negative. For example (-2)4 = 16 and (-2)5 = -32.
  • If 1 is raised to any power then the result will be always 1. For example, 13 = 1, 11001 = 1.
  • If any number except 1 is raised to power infinity then the result will be infinity. 2∞ = ∞

Laws of Exponents and Logarithms

The Laws of Exponents and the Logarithim Rules are two rules that are used to solve various mathematical problems and these rules are added in the table below.

Rules

Exponnets

Logarithms

Product Rule

xp.xq = x(p+q)

loga(mn) = logam + logan

Quotient Rule

xp/xq = x(p-q)

loga(m/n) = logam – logan

Power Rule

(xp)q = xp.q

logamn = n logam

Table: Laws of Exponents

The above-mentioned 7 Laws of Exponents are summarized in the following table:

Laws-of-Exponent

Also, Read

Exponent Rules Examples

Example 1: What is the simplification of 73 × 71?

Solution:

73 × 71 = 73+1 = 74

Example 2: Simplify and find the value of 102/52.

Solution:

We can write the given expression as;

102/52= (10/5)2 = 22 = 4

Example 3: Find the value of (256)3/4

Solution:

(256)3/4 = (44)3/4 = 44×(3/4) = 43 = 64

Example 4: Find the value of 7-3

Solution:

7-3 = (1/7)3 = 13/73 = 1/343

Example 5: Find the value of x if 125 = 25/5x

Solution: 

We have 125 = 25/5x

 ⇒ 53 = 52/5x

 ⇒ 53 = 52-x

Now the quantity is the same on both sides and bases are also the same, hence, exponents will also be the same. 

⇒ 3 = 2-x

⇒ x = 2-3 = -1

FAQs on Exponent Rules

1. What are Exponents in Maths?

Exponent refers to the power raised on a number which basically means the number is multiplied by itself to the number of times equal to the power.

2. What is the Product of Powers Rule?

Product of Power rule states that when two numbers with the same base are raised to different then the product of the number will have the power equal to the sum of the powers of both numbers. It is given as  xm × xn = x(m+n)

3. What is Power of Power Rule?

Power of Power rule states that when a number is raised to some power and the whole number including the first power is again raised to some power, then the two powers are multiplied.

4. What is Zero Exponent Rule?

Zero Exponent Rule states that if any number is raised to power 0 then it will result 1. It is given as X0 = 1.

5. What is the Value of 00?

The value of 00 is not defined in mathematics.

6. What are 8 Laws of Exponents?

The 8 Laws of Exponents are,

  • Product Law: am × an = am+n
  • Quotient Law: am/an = am-n
  • Zero Exponent Law: a0 = 1
  • Identity Exponent Law: a1 = a
  • Power of a Power: (am)n = amn
  • Power of a Product: (ab)m = ambm
  • Power of a Quotient: (a/b)m = am/bm
  • Negative Exponents Law: a-m = 1/am


Last Updated : 18 Feb, 2024
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