MULTIPLICATION WITH EXPONENTS

In this section, you will learn how to do binary operation multiplication between terms with exponents. 

We can use Products of Powers property to multiply terms with exponents. 

Products of Powers Property : 

Words : 

The product of two powers with the same base equals that base raised to the sum of the exponents. 

Numbers : 

5⁸ ⋅ 5⁴  =  5^{8 + 4}  =  5¹²

Algebra : 

If 'x' is any nonzero real number and 'm' and 'n' are integers, then, 

x^m ⋅ xⁿ  =  x^{m + n}

Example 1 :

Simplify : 

3⁴ ⋅ 3³

Solution : 

=  3⁴ ⋅ 3³

Product of Powers Property. 

=  3^{4 + 3}

=  3⁷

Example 2 :

Simplify : 

5^-5 ⋅ 5³

Solution : 

=  5^-5 ⋅ 5³

Product of Powers Property. 

=  5^{-5 + 3}

=  5^-2

=  1/5²

Example 3 :

Simplify : 

7⁶ ⋅ 7^-4

Solution : 

=  7⁶ ⋅ 7^-4

Product of Powers Property. 

=  7^{6 + (-4)}

=  7^{6 - 4}

=  7²

Example 4 :

Simplify : 

9^-2 ⋅ 9^-5

Solution : 

=  9^-2 ⋅ 9^-5

Product of Powers Property. 

=  9^{-2 + (-5)}

=  9^{-2 - 5}

=  9^-7

=  1/9⁷

Example 5 :

Simplify :

(-2)^-5 ⋅ (-2)⁶

Solution : 

=  (-2)^-5 ⋅ (-2)⁶

Product of Powers Property. 

=  (-2)^{-5 + 6}

 =  (-2)¹

  =  -2

Example 6 :

Simplify : 

(-4)⁵ ⋅ (-4)⁸

Solution : 

=  (-4)⁵ ⋅ (-4)⁸

Product of Powers Property. 

=  (-4)^{5 + 8}

=  (-4)¹³

Because the exponent is an odd number, negative sign inside the parentheses will remain same. 

=  -4¹³

Example 7 :

Simplify : 

(-7)⁹ ⋅ (-7)⁵

Solution : 

=  (-7)⁹ ⋅ (-7)⁵

Product of Powers Property. 

=  (-7)^{9 + 5}

=  (-7)¹⁴

Because the exponent is an even number, negative sign inside the parentheses will become positive.  

=  7¹⁴

Example 8 :

Simplify : 

4² ⋅ 3^-2 ⋅ 4⁵ ⋅ 3⁶

Solution : 

=  4² ⋅ 3^-2 ⋅ 4⁵ ⋅ 3⁶

Group powers with the same base together. 

=  (4² ⋅ 4⁵) ⋅ (3^-2 ⋅ 3⁶)

Product of Powers Property. 

=  4^{2 + 5} ⋅ 3^{-2 + 6}

=  4⁷ ⋅ 3⁴

Example 9 :

Simplify :

4P⁵ ⋅ 2P³ ⋅ P⁴ 

Solution : 

=  4P⁵ ⋅ 2P³ ⋅ P⁴

=  (4 ⋅ 2) ⋅ (P⁵ ⋅ P³ ⋅ P⁴)

Product of Powers Property. 

=  8 ⋅ P^{5 + 3 + 4}

=  8P¹²

Example 10 :

Find the value of x :  

(2/8)^2x  ⋅ (2/8)^x  =  (2/8)⁶

Solution : 

(2/8)^2x  ⋅ (2/8)^x  =  (2/8)⁶

Product of Powers Property. 

(2/8)^{2x + x}  =  (2/8)⁶

(2/8)^3x  =  (2/8)⁶

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

3x  =  6

Divide each side by 3.

x  =  2

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