APPLYING PROPERTIES OF INTEGER EXPONENTS

Properties of Exponents

Property 1 :

If two terms are multiplied with the same base, the base has to be taken once and exponents have to be added.  

That is,

x^m ⋅ xⁿ  =  x^m+n

Example :

3⁴ ⋅ 3⁵  =  3⁴⁺⁵

3⁴ ⋅ 3⁵  =  3⁹

Property 2 : 

If two terms are in division with the same base, the base has to be taken once and exponents have to be subtracted.

That is

x^m ÷ xⁿ  =  x^m-n

Example :

3⁷ ÷ 3⁵  =  3^7-5

3⁷ ÷ 3⁵  =  3²

Property 3 :

If there is an exponent for an exponential term, two exponents can either be multiplied or interchanged. 

That is

(x^m)ⁿ  =  x^mn

or 

(x^m)ⁿ  =  (xⁿ)^m

Example :

(3²)⁴  =  3⁽²⁾⁽⁴⁾

(3²)⁴  =  3⁸

Property 4 : 

If there is a common exponent for the product of two or more terms, the exponent can be distributed to each term.  

That is

(xy)^m  =  x^m ⋅ y^m

Example :

(3 ⋅ 5)²  =  3² ⋅ 5²

(3 ⋅ 5)²  =  9 ⋅ 25

(3 ⋅ 5)²  =  225.

Property 5 : 

If there is a common exponent for two terms in division, the exponent can be distributed to each term.  

That is

(x / y)^m  =  x^m / y^m

Example :

(3 / 5)²  =  3² / 5²

(3 / 5)²  =  9 / 25

Property 6 : 

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 7 :

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

That is

x⁰  =  1

Example :

3⁰  =  1

Property 8 :

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

That is

x  =  x¹

Example :

3¹  =  3

Solved Problems

Problem 1 :

Simplify the expression given below.

(5 - 2)⁵ · 3^-8 + (5 + 2)⁰

Solution : 

=  (5 - 2)⁵ · 3^-8 + (5 + 2)⁰

=  3⁵ · 3^-8 + 7⁰

=  3^5-8 + 1

=  3^-3 + 1

=  1/27 + 1

=  1/27 + 1

=  1/27 + 27/27

=  (1 + 27) / 27

=  28/27

Therefore, 

(5 - 2)⁵ · 3^-8 + (5 + 2)⁰  =  28/27

Problem 2 :

Simplify the expression given below.

[(3 + 1)²]³ / (7 - 3)²

Answer : 

=  [(3 + 1)²]³ / (7 - 3)²

=  [4²]³ / 4²

=  4⁶ / 4²

=  4^{6 - 2}

=  4⁴

=  4 ⋅ 4 ⋅ 4 ⋅ 4

=  256

Therefore, 

[(3 + 1)²]³ / (7 - 3)²  =  256

Problem 3 :

Simplify the expression given below.

[(6 - 1)²]² / (3 + 2)³

Answer : 

=  [(6 - 1)²]² / (3 + 2)³

=  [5²]² / 5³

=  5⁴ / 5³

=  5^{4 - 3}

=  5¹

=  5

Therefore, 

[(6 - 1)²]² / (3 + 2)³  =  5

Problem 4 :

Simplify the expression given below.

[(2)²]³ -  (10 - 6)⁴ ⋅ 4^-5

Answer : 

=  [(2)²]³ -  (10 - 6)⁴ ⋅ 4^-5

=  4³ -  4⁴ ⋅ 4^-5

=  4³ -  4^{4 - 5}

=  4³ -  4^-1

=  64 - 1/4

=  64 - 1/4

=  256/4 - 1/4

=  (256 - 1) / 4

=  255/4

Therefore, 

[(2)²]³ -  (10 - 6)⁴ ⋅ 4^-5  =  255/4

Problem 5 :

Simplify the expression given below.

[(8)²]^-1 ⋅ (10 - 8)¹⁴ ⋅ 4^-5

Answer : 

=  [(8)²]^-1 ⋅ (10 - 8)¹⁴ ⋅ 4^-5

=  (8)^-2 ⋅  2¹⁴ ⋅ (2²)^-5

=  (2³)^-2 ⋅  2¹⁴ ⋅ 2^-10

=  2^-6 ⋅  2¹⁴ ⋅ 2^-10

=  2^{-6 + 14 - 10}

=  2^-2

=  1/2²

=  1/4

Therefore, 

[(8)²]^-1 ⋅ (10 - 8)¹⁴ ⋅ 4^-5  =  1/4

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