Multiplying Powers with the Same Base but Different Powers: 

Exponents are also known as powers or indices. A power or exponent can be defined as the number of times a quantity is multiplied by itself. For example 4^ 2 ( four raised to the power of two) 

4^ 2= 4 * 4 = 16 

NOTE:  symbol ^ indicates raise to the power

When any two expressions with powers are multiplied it is known as multiplication exponents. If the multiplication is done with the same base  i.e. multiplying the powers with the same base, the following rules are to be taken into constopicIderation:

Rules of multiplying powers but having the same base

When the base of exponents has different powers 

x^a * x^b = x^ (a+b) 

When the base of exponents has the same power

x^a * y^a = (x*y)^a 

When both bass and exponents have different numbers

x^a * y^b = x^a * y^b

When the algebraic terms are multiplied, having the same base, the powers are added. Whenever multiplication of exponents is there, we need to add the powers. 

For example multiply x^2 * x^ 5 

                     = x^ ( 2+ 5) 

                     = x^ 7 

Here X raise to 2 is multiplied with x raised to 5. Then we will add the powers. 

The answer will be X raised to the power seven. 

 Example: multiply 5^3 * 5^ 4 

                = 5 ^ (3+4)

                = 5^7

Multiplication powers with different base and same powers: 

When in exponents, bases are different and powers are the same, we need to multiply the base and write the same power and evaluate the answer. 

Example : 10^ 3 * 5 ^3

                     = 10*5 ^ 3

                     = 50 ^ 3

                     = 125000

Multiplication powers with different base and powers:

When exponents, bases, and powers are different, we need to solve them indivtopicIdually and multiply their answers.  

Example: Multiply 6^2 * 2^3

              = (6*6) * (2*2*2) 

              = 36 * 8

              = 288 

Example 1: Solve 2^3 * 5^3 

= (2*5) ^3

= 10^ 3

= 10* 10 * 10

= 1000

Example 2: Multiply 2^7 * 2^4 * 2^7

= 2 ^ (7+4+7) 

= 2^18

Example 3 : Solve 5^0 * 6^2

= 1 * (6*6) 

= 1 * 36 

(as we know any number raised to 0 will always be equal to 1 it is as per the laws of exponent)

= 36

Multiplication of exponents with negative power

Whenever there is negative power to the base, we need to take reciprocal of the number and write positive exponents instead of negative power. 

Example: {3^(-2)} = {( 1/ 3) ^ 2} = 1/ 9 

Rules: 

when the bases are same but powers are different 

            x^-a * x^-b = x^ -(a+b) 

When powers are same but bases are different 

            x^-a * y^-a = (x*y)^ -a = ( 1/ x*y) ^a

When powers and bases both are different 

            x^a  * y^b

Example 4: multiply 4(^-3) *4(^-2) 

= 4^ -( 3+2)

= 4^ -5

= (1/ 4) ^5 = 1/256

Example 5: Solve 5(^ -2) * 6(^-2)

= (5* 6) ^ -2

= 30^-2 

= (1/30) ^ 2

= 1/900

Multiplication of fractional exponents:

Whenever there are fractional exponents to the base, we need to add the fractional exponents first and then we need to simplify the song.

Example: 4^½  * 4^⅓ 

= 4^(½ + ⅓ )

= 4^( ⅚)      here, we have found the LCM of two and three and have solved the fraction and we got the answer as ⅚

= 2^2*⅚ (First we have found the square root of four which is two. And then we have multiplied the power to with the power ⅚ .

= 2^ 5/3 (after simplification of 2×5/6 we got the answer as 5/3. 

Practice sums: 

Example 6:  Solve 49^2/7 * 49^1/7 

= 49^(2/7 +1/7)

=49^3/7

=7^2*3/7 

=7^6/7 

Example 7: Solve 256^¼   *  256 ^½ 

= 256^(¼ + ½) 

= 256^ ¾ 

= 2^8*¾ 

= 2^6

=64

Multiplication of variable exponents:

While multiplication of variable exponents, we need to see that we need to do the multiplication of numbers given and we need to add the like variables to get the answer and simplify it. 

Practice sums

Example 8: Solve 11a^3 b^2 * 5a^2 b^4

First, we will multiply both the numbers that are 11 and five separately. Then we will multiply like variables and use the law of exponents of multiplication in them.

= (11*5) * (a^3 * a^2)  * (b^2* b^4)

Now we will add the powers as whenever multiplication powers are there we add them for the same bases.

= 55 * (a^3+2)   * (b^2+4)

=55a^5 b^6  

Multiplication exponents with square root:

Now we will understand the multiplication of exponents where the basis has the square root. Here the exponent rule will remain the same if the bases are square roots. One point is to remember that we have to convert the radicals to rational exponents and then multiply the given expression. 

Example 9: Multiply (√3)^2 * (√5)^2

= 3 * 5  (Any square root raised to 2 will always be the same number like under root three raised to 2 equals to 3 same with under five raised to 2 equals to 5)

= 15 

Example 10: Solve √7 ^4  * √3^4 

Whenever the Power is the same then we can multiply the two under root together

= (√7* √3) ^4 

=√21^4