Powers and percentages

The powers

A power is a repeated multiplication:

2^5  =  2 x 2 x 2 x 2 x 2  =  32

This is read as ‘2 to the power of 5’. The 5 is called the exponent: it indicates the number of factors, not the number of multiplications.

The sign trap

(-3)^2  =  (-3) x (-3)  =  9
 -3^2   =  -(3 x 3)     =  -9

Without brackets, the power applies only to the 3, not to the minus sign. Two almost identical notations, two opposite results.

When multiplying two powers of the same number, we add the exponents:

2^3 x 2^4  =  2^7

This can be verified by counting: three factors of 2, then four factors of 2, which does indeed make seven factors of 2 in total.

When dividing, we subtract:

2^5 : 2^2  =  2^3

a^0   =  1        (for any value other than zero)
10^-3 =  1/1000  =  0,001

a^0 = 1 may seem surprising, but it is the only value consistent with the previous rule: 2^3 : 2^3 equals 1, and so does 2^0.

We use them every day to write both large and small numbers:

The exponent counts the zeros. This is what makes 10^9 much easier to read than 1,000,000,000.