1.2 Learn Basic Properties and Operations

The powers of ten form patterns that can help you understand exponents. In the examples, pay close attention to the pattern formed by the zeros.

Examples

Did you notice? The powers of ten are very easy to figure out. Each is equal to 1 followed by the number of zeros shown by the exponent.

This pattern also reveals what will happen when we raise a number to the first or zero power.

Any number raised to a power of 1 doesn’t change: $${a^1} = a$$
Any number raised to a power of 0 equals 1: $${a^0} = 1$$

Exponents can also be negative numbers. You already know that to divide exponents, you should subtract.

Examples

$${\textstyle{{{8^3}} \over {{8^5}}}} = {8^{3 - 5}} = {8^{ - 2}}$$

You can also think of it this way:

You can see that $${8^{ - 2}} = {\textstyle{1 \over {{8^2}}}}$$ . When you see a negative exponent, move the base and its exponent to the other side of the fraction bar and change the sign of the exponent.

Examples