# 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

xm ⋅ xn  =  xm+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

(xm)n  =  xmn

Example :

(32)4  =  3 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  =  xm ⋅ ym

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

xm ÷ xn  =  xm-n

Example :

37 ÷ 35  =  37-5

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  =  xm/ym

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  =  ym / xm

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/xm

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

xm/n  =  y -----> x  =  yn/m

Example :

x1/2  =  3

x  =  32/1

x  =  32

x  =  9

Property 5 :

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

That is

ax  =  ay -----> x  =  y

Example :

3m  =  35 -----> m  =  3

Property 6 :

If two powers 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 above, if you need any other stuff in math, please use our google custom search here.

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