Any non-zero base raised to the power of zero is equal to one.
Formula:
$a^0 = 1$, where $a \neq 0$
Important Note: $0^0$ is undefined.
Examples:
$5^0 = 1$
$100^0 = 1$
$30^0 = 1$
Example: Express $432$ as a product of powers of prime factors.
Solution:
In power notation:
Example: Compare $2.7 \times 10^{12}$ and $1.5 \times 10^8$.
Solution:
To compare, rewrite $2.7 \times 10^{12}$ to have the same power of 10 as the other term:
Since $27000 > 1.5$, we conclude that:
If $a^m = a^n$, then $m=n$, where $a \neq -1, 0, 1$.
Example: If $2^{x+7} = 128$, find $x$.
Solution:
Express 128 as a power of 2: $128 = 2^7$.
Set the exponents equal: $x+7 = 7$.
Solve for $x$: $x = 7 – 7 = 0$.