PROPERTIES OF EXPONENTS

Product of Powers Property :

The product of two powers with the same base equals that base raised to the sum of the exponents. 

If x is any nonzero real number and m and n are integers, then

x^m ⋅ xⁿ  =  x^m+n

Example :

3⁴ ⋅ 3⁵  =  3⁴⁺⁵

3⁴ ⋅ 3⁵  =  3⁹

Power of a Power Property :

A power raised to another power equals that base raised to the product of the exponents. 

If x is any nonzero real number and m and n are integers, then

(x^m)ⁿ  =  x^mn

Example :

(3²)⁴  =  3^2 ⋅ 4

(3²)⁴  =  3⁸

Power of a Product Property : 

A product raised to a power equals the product of each factor raised to that power.  

If x and y are any nonzero real numbers and m is any integer, then

(xy)^m  =  x^m ⋅ y^m

Example :

(3 ⋅ 5)²  =  3² ⋅ 5²

(3 ⋅ 5)²  =  9 ⋅ 25

(3 ⋅ 5)²  =  225.

Quotient of Powers Property : 

The quotient of two non zero powers with the same base equals the base raised to the difference of the exponents. 

If x is any nonzero real number and m and n are integers, then

x^m ÷ xⁿ  =  x^m-n

Example :

3⁷ ÷ 3⁵  =  3^7-5

3⁷ ÷ 3⁵  =  3²

Positive Power of a Quotient Property : 

A quotient raised to a positive power equals the quotient of each base raised to that power. 

If x and y are any nonzero real numbers and m is a positive integer, then

(x/y)^m  =  x^m/y^m

Example :

(3/5)²  =  3²/5²  =  9/25

Negative Power of a Quotient Property : 

A quotient raised to a negative power equals the reciprocal of the quotient raised to the opposite (positive) power. 

If x and y are any nonzero real numbers and m is a positive integer, then

(x / y)^-m  = (y / x)^m  =  y^m / x^m

Example :

(3/2)^-2  =  (2/3)²  =  2²/3²  =  4/9

Some More Properties of Exponents

Property 1 : 

If a term is moved from numerator to denominator or denominator to numerator, the sign of the exponent has to be changed. 

That is

x^-m  =  1/x^m

Example :

3^-2  =  1/3²

3^-2  =  1/9

Property 2 :

For any nonzero base, if the exponent is zero, its value is 1. 

That is

x⁰  =  1

Example :

3⁰  =  1

Property 3 :

For any base base, if there is no exponent, the exponent is assumed to be 1.

That is

x  =  x¹

Example :

3¹  =  3

Property 4 :

If an exponent is transferred from one side of the equation to the other side of the equation, reciprocal of the exponent has to be taken. 

That is

x^m/n  =  y -----> x  =  y^n/m

Example :

x^1/2  =  3

x  =  3^2/1

x  =  3²

x  =  9

Property 5 :

If two powers are equal with the same base, exponents can be equated.   

That is

a^x  =  a^y -----> x  =  y

Example :

3^m  =  3⁵ -----> m  =  3

Property 6 :

If two powers are equal with the same exponent, bases can be equated.   

That is

x^a  =  y^a -----> x  =  y

Example :

k³  =  5³ -----> k  =  5

Practice Problems

Problem 1 :

Simplify : 

2m² ⋅ 2m³

Solution :

2m² ⋅ 2m³  =  2m² ⋅ 2m³

2m² ⋅ 2m³  =  4m⁽²⁺³⁾

2m² ⋅ 2m³  =  4m⁵

Problem 2 :

Simplify :

m⁴ ⋅ 2m^-3

Solution :

m⁴ ⋅ 2m^-3  =  m⁴ ⋅ 2m^-3

m⁴ ⋅ 2m^-3  =  2m^{(4 - 3)}

m⁴ ⋅ 2m^-3  =  2m¹

m⁴ ⋅ 2m^-3  =  2m

Problem 3 :

Simplify :

(4a³)²

Solution :

(4a³)²  =  4²(a³)²

(4a³)²  =  16a⁽³⁾⁽²⁾

(4a³)²  =  16a⁶

Problem 4 :

Simplify : 

(x³)⁰

Solution :

(x³)⁰  =  1

Problem 5 :

Simplify : 

(12a³b²) / (3a⁴b³)

Solution :

(12a³b²) / (3a⁴b³)  =  (12/3)a^3-4b^2-3

(12a³b²) / (3a⁴b³)  =  4a^-1b^-1

(12a³b²) / (3a⁴b³)  =  4 / (a¹b¹)

(12a³b²) / (3a⁴b³)  =  4 / (ab)

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