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 : **

5^{8} ⋅ 5^{4} = 5^{8 + 4} = 5^{12}

**Algebra : **

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

x^{m} ⋅ x^{n} = x^{m + n}

**Example 1 :**

Simplify :

3^{4} ⋅ 3^{3}

**Solution : **

= 3^{4} ⋅ 3^{3}

Product of Powers Property.

= 3^{4 + }^{3}

= 3^{7}

**Example 2 :**

Simplify :

5^{-5} ⋅ 5^{3}

**Solution : **

= 5^{-5} ⋅ 5^{3}

Product of Powers Property.

= 5^{-5 + }^{3}

= 5^{-2}

= 1/5^{2}

**Example 3 :**

Simplify :

7^{6} ⋅ 7^{-4}

**Solution : **

= 7^{6} ⋅ 7^{-4}

Product of Powers Property.

= 7^{6 + (-4)}

= 7^{6 - 4}

= 7^{2}

**Example 4 :**

Simplify :

9^{-2} ⋅ 9^{-5}

**Solution : **

= 9^{-2} ⋅ 9^{-5}

Product of Powers Property.

= 9^{-2 + (-}^{5)}

= 9^{-2 - }^{5}

= 9^{-7}

= 1/9^{7}

**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.

= -4^{13}

**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.

= 7^{14}

**Example 8 :**

Simplify :

4^{2} ⋅ 3^{-2 }⋅ 4^{5 }⋅ 3^{6}

**Solution : **

= 4^{2} ⋅ 3^{-2 }⋅ 4^{5 }⋅ 3^{6}

Group powers with the same base together.

= (4^{2} ⋅ 4^{5}) ⋅ (3^{-2 }⋅ 3^{6})

Product of Powers Property.

= 4^{2 + }^{5} ⋅ 3^{-2 + 6}

= 4^{7} ⋅ 3^{4}

**Example 9 :**

Simplify :

4P^{5 }⋅ 2P^{3} ⋅ P^{4}

**Solution : **

= 4P^{5 }⋅ 2P^{3} ⋅ P^{4}

= (4 ⋅ 2) ⋅ (P^{5 }⋅ P^{3} ⋅ P^{4})

Product of Powers Property.

= 8 ⋅ P^{5 + }^{3 + }^{4}

= 8P^{12}

**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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