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Applying the Properties of Exponents

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- Properties of Exponents
- Cautions when Applying the Properties
- Applying the Properties of Exponents

**Properties of Exponents**

There are several properties of exponents involving multiplication, division, and powers that can help us simplify expressions containing exponents. They are:

The Product Power:

The Quotient Power:

The Power of Powers:

Power of a Product:

Power of a Quotient:

Negative Exponents:

**Cautions with Applying these Properties**

There are a couple of things to watch out for when applying these properties. First, it is important to remember that the product and quotient powers can only be applied with the bases are the same. We cannot simply add the exponents if the bases are not the same.

Secondly, the product and quotient of power properties only work with multiplication and division; they do not work with addition or subtraction.

**Applying the Properties of Exponents**

Let's go through some examples where we can use different properties of exponents to simplify expressions:

EXAMPLE 1

If we see factors that appear in both the numerator and denominator, we cancel them out. This means that we reduce the power in both the numerator and denominator. If we cancel all factors in the denominator, the denominator is 1, and doesn't need to be written.

EXAMPLE 2