LAWS OF EXPONENTS

Law 1 : 

x^m ⋅ xⁿ  =  x^m+n

Example :

3⁴ ⋅ 3⁵  =  3⁴⁺⁵

3⁴ ⋅ 3⁵  =  3⁹

Law 2 : 

x^m ÷ xⁿ  =  x^m-n

Example :

3⁷ ÷ 3⁵  =  3^7-5

3⁷ ÷ 3⁵  =  3²

Law 3 : 

(x^m)ⁿ  =  x^mn

Example :

(3²)⁴  =  3⁽²⁾⁽⁴⁾

(3²)⁴  =  3⁸

Law 4 : 

(xy)^m  =  x^m ⋅ y^m

Example :

(3 ⋅ 5)²  =  3² ⋅ 5²

(3 ⋅ 5)²  =  9 ⋅ 25

(3 ⋅ 5)²  =  225.

Law 5 : 

(x / y)^m  =  x^m / y^m

Example :

(3 / 5)²  =  3² / 5²

(3 / 5)²  =  9 / 25

Law 6 : 

x^-m  =  1 / x^m

Example :

3^-2  =  1 / 3²

3^-2  =  1 / 9

Law 7 : 

x⁰  =  1

Example :

3⁰  =  1

Law 8 : 

x¹  =  x

Example :

3¹  =  3

Law 9 : 

x^m/n  =  y -----> x  =  y^n/m

Example :

x^1/2  =  3

x  =  3^2/1

x  =  3²

x  =  9

Law 10 : 

(x / y)^-m  =  (y / x)^m

Example :

(5 / 3)^-2  =  (3 / 5)²

(5 / 3)^-2  =  3² /  5²

(5 / 3)^-2  =  9 / 25

Law 11 : 

a^x  =  a^y -----> x  =  y

Example :

3^m  =  3⁵ -----> m  =  5

Law 12 : 

x^a  =  y^a -----> 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 n  =  1² + 1⁴ + 1⁶ + 1⁸ +...............+ 1⁵⁰, then find the value of n. 

Solution : 

1 to the power of anything is equal to 1.

So, we have

1²  =  1

1⁴  =  1

1⁶  =  1

and so on. 

List out all the exponents. 

2, 4, 6, 8, ................ 48, 50

Divide each element by 2.

1, 2, 3, 4, ................ 24, 25

We can clearly see there are 25 terms in the series shown below. 

1² + 1⁴ + 1⁶ + 1⁸ +...............+ 1⁵⁰

Therefore, n is the sum of twenty-five 1's. 

That is, 

n  =  1² + 1⁴ + 1⁶ + 1⁸ +...............+ 1⁵⁰

n  =  25

Problem 3 : 

If 4^{2n + 3}  =  8^{n + 5}, then find the value of n. 

Solution : 

4^{2n + 3}  =  8^{n + 5}

(2²)^{2n + 3}  =  (2³)^{n + 5}

2^{2(2n + 3)}  =  2^{3(n + 5)}

Equate the exponents. 

2(2n + 3)  =  3(n + 5)

4n + 6  =  3n + 15

n  =  9

Problem 4 : 

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

Solution : 

2^x / 2^y  =  2³

2^{x - y}  =  2³

x - y  =  3

x  =  y + 3

Problem 5 : 

If a^x = b, b^y = c and  c^z = a, then find the value of xyz. 

Solution : 

Let

a^x  =  b -----(1)

b^y  =  c -----(2)

c^z  =  a -----(3)

Substitute a  =  c^z in (1).

(1)-----> (c^z)^x  =  b

c^zx  =  b

Substitute c  =  b^y.

(b^y)^zx  =  b

b^xyz  =  b

b^xyz  =  b¹

xyz  =  1

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