CHANGING THE BASE OF AN EXPONENT

Example 1 :

Rewrite 9³⁰⁰⁰ with base 3.

Solution :

=  9³⁰⁰⁰

=  (3²)³⁰⁰⁰

Power of a Power Property. 

=  3^2 ⋅ 3000

=  3⁶⁰⁰⁰

Example 2 :

Rewrite 27⁴⁰⁰⁰ with base 3.

Solution :

=  27⁴⁰⁰⁰

=  (3³)⁴⁰⁰⁰

Power of a Power Property. 

=  3^{3 ⋅ 4000}

=  3¹²⁰⁰⁰

Example 3 :

Rewrite 5⁴⁰⁰⁰ with base 25.

Solution :

=  5⁴⁰⁰⁰

=  5^{2 ⋅ 2000}

Power of a Power Property. 

=  (5²)²⁰⁰⁰

=  25²⁰⁰⁰

Example 4 :

Rewrite 2³⁰⁰⁰ with base 8.

Solution :

=  2³⁰⁰⁰

=  2^{3 ⋅ 1000}

Power of a Power Property. 

=  (2³)¹⁰⁰⁰

=  8¹⁰⁰⁰

Example 5 :

Rewrite 2⁵ ⋅ 8¹⁰⁰⁰ with base 2.

Solution :

=  2⁵ ⋅ 8¹⁰⁰⁰

=  2⁵ ⋅ (2³)¹⁰⁰⁰

Power of a Power Property. 

=  2⁵ ⋅ 2^{3 ⋅ 1000}

=  2⁵ ⋅ 2³⁰⁰⁰

Product of Powers Property. 

=  2^{3000 + 5}

=  2³⁰⁰⁵

Example 6 :

Rewrite 5³ ⋅ 25²⁰⁰⁰ with base 5.

Solution :

=  5³ ⋅ 25²⁰⁰⁰

=  5³ ⋅ (5²)²⁰⁰⁰

Power of a Power Property. 

=  5³ ⋅ 5^{2 ⋅ 2000}

=  5³ ⋅ 5⁴⁰⁰⁰

Product of Powers Property. 

=  5^{3 + 4000}

=  5⁴⁰⁰³

Example 7 :

Rewrite 2¹⁰⁰ · 4²⁰⁰ · 8³⁰⁰ with base 2.

Solution :

=  2¹⁰⁰ · 4²⁰⁰ · 8³⁰⁰

=  2¹⁰⁰ · (2²)²⁰⁰ · (2³)²⁰⁰

Power of a Power Property. 

=  2¹⁰⁰ · 2⁴⁰⁰ · 2⁶⁰⁰

Product of Powers Property. 

=  2^{100 + 400 + 600}

=  2¹¹⁰⁰

Example 8 :

Rewrite 3⁵⁰⁰ · 9²⁰⁰ · 27¹⁰⁰ with base 3.

Solution :

=  3⁵⁰⁰ · 9²⁰⁰ · 27¹⁰⁰

=  3⁵⁰⁰ · (3²)²⁰⁰ · (3³)¹⁰⁰

Power of a Power Property. 

=  3⁵⁰⁰ · 3⁴⁰⁰ · 3³⁰⁰

Product of Powers Property. 

=  3^{(500 + 400 + 300)}

=  3¹²⁰⁰

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