# 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,

xm ⋅ xn  =  xm+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

xm ÷ xn  =  xm-n

Example :

37 ÷ 35  =  37-5

37 ÷ 35  =  32

Property 3 :

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

That is

(xm)n  =  xmn

or

(xm)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  =  xm ⋅ ym

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  =  xm / ym

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 / xm

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

Property 9 :

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

xm/n  =  y -----> x  =  yn/m

Example :

x1/2  =  3

x  =  32/1

x  =  32

x  =  9

Property 10 :

If the sign of the exponent is changed, reciprocal has to be taken for the base.

That is

(x / y)-m  =  (y / x)m

Example :

(5 / 3)-2  =  (3 / 5)2

(5 / 3)-2  =  32 /  52

(5 / 3)-2  =  9 / 25

Property 11 :

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

That is

ax  =  ay -----> x  =  y

Example :

3m  =  35 -----> m  =  3

Property 12 :

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

That is

xa  =  ya -----> 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) =  42(a3)2

(4a3)2  =  16a(3)(2)

(4a3)2  =  16a6

Problem 4 :

Simplify :

(x3)0

Solution :

(x3) =  1

Problem 5 :

Simplify :

(12a3b2) / (3a4b3)

Solution :

(12a3b2) / (3a4b3)  =  (12/3)a3-4b2-3

(12a3b2) / (3a4b3)  =  4a-1b-1

(12a3b2) / (3a4b3)  =  4 / (a1b1)

(12a3b2) / (3a4b3)  =  4 / (ab) Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

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