Simplification of powers

To simplify powers means to use the rules of powers to write an expression in a simpler form. Powers are used to make repeated multiplication shorter, and with simplification we can rewrite and collect expressions.

 

Example 1: Same base with multiplication

When we multiply powers with the same base, we add the exponents:

 

$$ \large a^3 \cdot a^4 = a^{3+4} = a^7 $$

 

 

Example 2: Same base with division

When we divide powers with the same base, we subtract the exponents:

 

$$ \large \tfrac{a^7}{a^3} = a^{7-3} = a^4 $$

 

 

Example 3: Power of a power

When we have a power raised to another power, we multiply the exponents:

 

$$ \large (a^3)^4 = a^{3 \cdot 4} = a^{12} $$

 

Example 4: Several different bases

When we multiply different bases, we cannot add the exponents, but we can write it together:

 

$$ \large 2^3 \cdot 5^3 = (2 \cdot 5)^3 = 10^3 $$

 

 

Example 5: Negative exponents

A negative exponent means that we take the reciprocal value:

 

$$ \large a^{-3} = \tfrac{1}{a^3} $$

 

 

Example 6: Zero as an exponent

Any number (except 0) raised to the power of 0 always equals 1:

 

$$ \large a^0 = 1 \quad (a \neq 0) $$

 

 

Example 7: Combination of rules

Simplify the expression:

 

$$ \large \tfrac{a^5 \cdot a^3}{a^4} $$

 

First we collect in the numerator:

 

$$ \large a^{5+3} = a^8 $$

 

Then we divide:

 

$$ \large \tfrac{a^8}{a^4} = a^{8-4} = a^4 $$

 

 

Summary

When you simplify powers, remember:

 

• $$ \large a^m \cdot a^n = a^{m+n} $$
• $$ \large \tfrac{a^m}{a^n} = a^{m-n} $$
• $$ \large (a^m)^n = a^{m \cdot n} $$
• $$ \large a^0 = 1 \quad (a \neq 0) $$
• $$ \large a^{-n} = \tfrac{1}{a^n} $$

 

Simplification of powers makes expressions simpler and is an important part of working with algebra, equations and functions.