Using Exponents in Algebra

You might like to read the page on Exponents first.

Whole Number Exponents

The exponent "n" in aⁿ says how many times to use a in a multiplication:

The "3" says to use 5 three times in a multiplication
In words: 5³ could be called "5 to the third power", "5 to the power 3" or simply "5 cubed"

negative-exponent

A negative exponent a⁻ⁿ is equal to 1/aⁿ (1 divided by the positive exponent)

5⁻³ = 1/5³ = 1/125 = 0.008

Also ... by changing the signs of the exponents we get:

negative-exponent

A positive exponent aⁿ is equal to 1/a⁻ⁿ (1 divided by the negative exponent)

So, we can move an expression between the top and bottom (numerator and denominator) of a fraction by simply changing the sign of the exponent.

Example: x⁻¹ = 1/x (a simple reciprocal)

Example: 5⁻³ = 1/5³ = 1/125 = 0.008

Here is an example with positive and negative exponents:

We can put the negative exponents at the bottom (and they become positive):

4³3²2⁵9

3² is 9:

4³92⁵9

The 9s cancel out:

4³2⁵

4³ is 4 × 4 × 4 = 64, and 2⁵ is 2 × 2 × 2 × 2 × 2 = 32:

6432

Which simplifies to:

Done!