SIMPLIFYING EXPONENTIAL EXPRESSIONS

Key Concept

An exponential expression is completely simplified, if.......

• There are no negative exponents. 

• The same base does not appear more than once in a product or quotient. 

• No powers are raised to powers. 

• No products are raised to powers. 

• No quotients are raised to powers. 

• Numerical coefficients in a quotient do not have any common factor other than 1. 

Finding Product of Powers

Example 1 :

Simplify : 

a⁴ ⋅ b⁵ ⋅ a²

Solution :

=  a⁴ ⋅ b⁵ ⋅ a²

Group powers with the same base together. 

=  (a⁴ ⋅ a²) ⋅ b⁵

Add the exponents of powers with the same base.   

=  a^{4 + 2} ⋅ b⁵

=  a⁶ ⋅ b⁵

Example 2 :

Simplify : 

x² ⋅ x ⋅ x^-4

Solution :

=  x² ⋅ x ⋅ x^-4

Because the powers have the same base, keep the base and add the exponents.  

=  x^{2 + 1 + (-4)}

=  x^{2 + 1 - 4}

=  x^-1

=  1/x

Finding Powers of Powers

Example 3 :

Simplify : 

(y²)^-4 ⋅ y⁵

Solution :

=  (y²)^-4 ⋅ y⁵

Use the Power of a Power Property. 

=  y^-8 ⋅ y⁵

=  y^{-8 + 5}

=  y^-3

=  1/y³

Example 4 :

Simplify : 

(xy²)^-4 ⋅ (x³y⁵)²

Solution :

=  (xy²)^-4 ⋅ (x³y⁵)²

Use the Power of a Power Property. 

=  x^-4(y²)^-4 ⋅ (x³)²(y⁵)²

=  x^-4y^-8 ⋅ x⁶y¹⁰

Group powers with the same base together. 

=  (x^-4 ⋅ x⁶) ⋅ (y^-8 ⋅ y¹⁰)

=  x^{-4 + 6} ⋅ y^{-8 + 10}

=  x²y²

Finding Powers of Product

Example 5 :

Simplify : 

(-5x)²

Solution :

=  (-5x)²

Use the Power of a Product Property. 

=  (-5)² ⋅ x²

=  25x²

Example 6 :

Simplify : 

-(5x)²

Solution :

=  -(5x)²

Use the Power of a Product Property. 

=  -(5² ⋅ x²)

=  -(25 ⋅ x²)

=  -25x²

Example 7 :

Simplify : 

(a^-2 ⋅ b⁰)³

Solution :

=  (a^-2 ⋅ b⁰)³

Use the Power of a Product Property. 

=  (a^-2)³ ⋅ (b⁰)³

=  a^-2 ⋅ 3 ⋅ b^0 ⋅ 3

=  a^-6 ⋅ b⁰

=  x^-6 ⋅ 1

=  1/a⁶

Finding Quotient of Powers

Example 8 :

Simplify : 

xy³/y⁵

Solution :

=  xy³/y⁵

Use the Quotient of Powers Property. 

=  xy^{3 - 5}

=  xy^-2

=  x/y²

Example 9 :

Simplify : 

x⁵y⁹/(xy)⁴

Solution :

=  x⁵y⁹/(xy)⁴

Use the Power of a Product Property. 

=  x⁵y⁹/x⁴y⁴

Use the Quotient of Powers Property. 

=  x^5-4 ⋅ y^9-4

=  x¹ ⋅ y⁵

=  xy⁵

Finding Positive Powers of Quotients

Example 10 :

Simplify : 

(2a³/bc)³

Solution :

=  (2a³/bc)³

Use the Power of a Quotient Property. 

=  (2a³)³/(bc)³

Use the Power of a Power Property. 

=  2³(a³)³/(b³c³)

=  8a⁹/b³c³

Finding Negative Powers of Quotients

Example 11 :

Simplify : 

(3a/b²)^-3

Solution :

=  (3a/b²)^-3

Rewrite with a positive exponent. 

=  (b²/3a)³

Use the Power of a Quotient Property. 

=  (b²)³/(3a)³

Use the Power of a Power Property. 

=  b⁶/(3³a³)

=  b⁶/27a³

Example 12 :

Simplify : 

(3/4)^-1 ⋅ (2a/3b)^-2

Solution :

=  (3/4)^-1 ⋅ (2a/3b)^-2

Rewrite each fraction with a positive exponent. 

=  (4/3)¹ ⋅ (3b/2a)²

Use the Power of a Quotient Property. 

=  (4/3) ⋅ (3b)²/(2a)²

Use the Power of a Power Property. 

=  (4/3) ⋅ (3²b²/2²a²)

=  (4/3) ⋅ (9b²/4a²)

Simplify. 

=  3b²/a²

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