DIVISION PROPERTIES OF EXPONENTS

Quotient of Powers Property

Words :

The quotient of two non zero powers with the same base equals the base raised to the difference of the exponents. 

Numbers :

3⁷ ÷ 3⁵  =  3^{7 - 5}  =  3²

Algebra :

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

x^m ÷ xⁿ  =  x^{m - n}

Finding Quotient of Powers

Example 1 :

Simplify : 

3⁸/3²

Solution :

=  3⁸/3²

Use the Quotient of Powers Property. 

=  3^{8 - 2}

=  3⁶

=  729

Example 2 :

Simplify : 

y³/y³

Solution :

=  y³/y³

Use the Quotient of Powers Property. 

=  y^{3 - 3}

=  y⁰

=  1

Example 3 :

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

Example 4 :

Simplify : 

(4³ ⋅ 5² ⋅ 6⁷)/(4 ⋅ 5⁴ ⋅ 6⁵)

Solution :

=  (4³ ⋅ 5² ⋅ 6⁷)/(4 ⋅ 5⁴ ⋅ 6⁵)

Use the Quotient of Powers Property. 

=  4^3-1 ⋅ 5^2-4 ⋅ 6^7-5

=  4² ⋅ 5^-2 ⋅ 6²

=  (4² ⋅ 6²)/5²

=  (16 ⋅ 36)/25

=  576/25

Dividing Numbers in Scientific Notation

Example 5 :

Simplify (4 x 10⁹) ÷ (16 x 10⁶) and write the answer in scientific notation.  

Solution :

=  (4 x 10⁹) ÷ (16 x 10⁶)

=  (4 x 10⁹)/(16 x 10⁶)

Write as a product of quotients. 

=  (4/16) x (10⁹/10⁶)

Simplify each quotient. 

=  0.25 x 10^{9 - 6}

=  0.25 x 10³

Write 0.25 in scientific notation as 2.5 x 10^-1. 

=  2.5 x 10^-1 x 10³

The second two terms have the same base, so add the exponents.  

=  2.5 x 10^{-1 + 3}

=  2.5 x 10²

Positive Power of a Quotient Property

Words :

A quotient raised to a positive power equals the quotient of each base raised to that power. 

Numbers :

(5/7)²  =  5²/7²  =  25/49

Algebra :

If x and y are any nonzero real numbers and m is a positive integer, then

(x/y)^m  =  x^m/y^m

Finding Positive Powers of Quotients

Example 6 :

Simplify : 

(2/5)³

Solution :

=  (2/5)³

Use the Power of a Quotient Property. 

=  2³/5³

=  8/125

Example 7 :

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

Negative Power of a Quotient Property

Words :

A quotient raised to a negative power equals the reciprocal of the quotient raised to the opposite (positive) power. 

Numbers :

(3/2)^-2  =  (2/3)²  =  2²/3²  =  4/9

Algebra :

If x and y are any nonzero real numbers and m is a positive integer, then

(x/y)^-m  =  (y/x)^m  =  y^m/x^m

Finding Negative Powers of Quotients

Example 8 :

Simplify : 

(5/4)^-3

Solution :

=  (5/4)^-3

Rewrite with a positive exponent. 

=  (4/5)³

Use the Power of a Quotient Property. 

=  4³/5³

=  64/125

Example 9 :

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 10 :

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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