# Raising a Product or Quotient to a Power

It is important to know some general rules about exponents:

\begin{align*}\text{Raising a product to a power:} && (x \cdot y)^n & = x^n \cdot y^n\\ \text{Raising a quotient to a power:} && \left(\frac{x}{y} \right)^n & = \frac{x^n}{y^n}\end{align*}

In radical notation, these properties are written as

\begin{align*}\text{Raising a product to a power:} && \sqrt[m]{x \cdot y} & = \sqrt[m]{x} \ \cdot \sqrt[m]{y}\\ \text{Raising a quotient to a power:} && \sqrt[m]{\frac{x}{y}} & = \frac{\sqrt[m]{x}}{\sqrt[m]{y}}\end{align*}

####
**Exponent Rule #1**

*Whenever you multiply terms of the same base, you can add the exponents.*

**( x ^{m} ) ( x ^{n} ) = x^{( m + n )}**

**Note:** You ** cannot ** add the exponents in this case: (

*x*

^{4})(

*y*

^{3})

**because x and y are different base terms.**

Practice:

- Simplify\begin{align*}(x^3)(x^4)\end{align*}
- Simplify \begin{align*}(q^6)(q^-2)\end{align*}

**Exponent Rule #2**

Whenever you have an exponent expression that is raised to a power, you can multiply the exponent and power:

**( x^{m} ) ^{n} = x ^{m n}**

Practice:

- Simplify \begin{align*}(x^5)^6\end{align*}
- Simplify \begin{align*}(x^7)^3\end{align*}
- Simplify \begin{align*}(xy^2)^3\end{align*} (Note the "squared" applies to both x and y).

**Exponent Rule #3**

Anything to the power of zero is one.

\begin{align*}x^0=1\end{align*}

**Exponent Rule #4**

If you are dividing quantities with the same base term, you can subtract the exponents.

Practice:

- \begin{align*}Simplify: \frac{x^4}{x^3}\end{align*}
- \begin{align*}Simplify: \frac{x^-6}{x^4}\end{align*}

**Exponent Rule #5**

Practice:

- \begin{align*}Simplify: (\frac{x}{y})^3\end{align*}

These are the basic rules, although there are many more laws of exponents that stem from these.