# EXPONENT RULES

Rule 1 :

If two powers are multiplied with the same base, then the base has to be taken once and raised to sum of the exponents

xm ⋅ xn  =  xm+n

Example :

Rule 2 :

If two powers are divided with the same base, then the base has to be taken once and raised to difference of the exponents

xm ÷ xn  =  xm-n

Example :

Rule 3 :

If a power raised to another exponent, then the two exponents can be multiplied.

(xm)n  =  xmn

Example :

Rule 4 :

If there is a common exponent for the product of two different values, then the exponent can be distributed to each value and the results are multiplied.

(xy)m  =  xm ⋅ ym

Example :

(3 ⋅ 5)2  =  32 ⋅ 52

=  9 ⋅ 25

=  225.

Rule 5 :

If there is a common exponent for the division of two different values, then the exponent can be distributed to each value and the results are divided.

(x/y)m  =  xm/ym

Example :

(3/5)2  =  32/52

=  9/25

Rule 6 :

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

x-m  =  1/xm

Example :

3-2  =  1/32

=  1/9

Rule 7 :

If the exponent is zero for any non zero base, its value is equal to 1.

x0  =  1

Example :

30  =  1

Rule 8 :

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

x1  =  x

Example :

31  =  3

Rule 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.

xm/n  =  y -----> x  =  yn/m

Example :

x1/2  =  3

x  =  32/1

x  =  32

x  =  9

Rule 10 :

Any non zero base raised to a negative exponent equals the reciprocal of the base raised to the opposite (positive) of the exponent.

(x/y)-m  =  (y/x)m

Example :

(5/3)-2  =  (3/5)2

=  32/52

=  9/25

Rule 11 :

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

ax  =  ay -----> x  =  y

Example :

3m  =  35 -----> m  =  3

Rule 12 :

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

xa  =  ya -----> x  =  y

Example :

k3  =  53 -----> k  =  5

## Important Note

Many students do not know the difference between

(-3)2   and   -32

Order of operations (PEMDAS) dictates that parentheses take precedence.

So, we have

(-3)2  =  (-3) ⋅ (-3)

(-3)2  =  9

Without parentheses, exponents take precedence :

-32  =  -3 ⋅ 3

-32  =  -9

The negative is not applied until the exponent operation is carried through. We have to make sure that we understand this. So, we will not make this common mistake.

Sometimes, the result turns out to be the same, as in.

(-2)3   and   -23

We have to make sure why they yield the same result.

## Practice Problems

Problem 1 :

If a-1/2  =  5, then find the value of a.

Solution :

a-1/2  =  5

a  =  5-2/1

a  =  5-2

a  =  1/52

a  =  1/25

Problem 2 :

If 42n + 3  =  8n + 5, then find the value of n.

Solution :

42n + 3  =  8n + 5

(22)2n + 3  =  (23)n + 5

22(2n + 3)  =  23(n + 5)

Equate the exponents.

2(2n + 3)  =  3(n + 5)

4n + 6  =  3n + 15

n  =  9

Problem 3 :

If 2x / 2y  =  23, then find the value x in terms of y.

Solution :

2x / 2y  =  23

2x - y  =  23

x - y  =  3

x  =  y + 3

Problem 4 :

If ax = b, by = c and  cz = a, then find the value of xyz.

Solution :

Let

ax  =  b -----(1)

by  =  c -----(2)

cz  =  a -----(3)

Substitute a  =  cin (1).

(1)-----> (cz)x  =  b

czx  =  b

Substitute c  =  by.

(by)zx  =  b

bxyz  =  b

bxyz  =  b1

xyz  =  1

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