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Raising a Product or Quotient to a Power

Exponents under radicals

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Raising a Product or Quotient to a Power

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: (x4)(y3) because x and y are different base terms.

Practice:

  1. Simplify\begin{align*}(x^3)(x^4)\end{align*}
  2. 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:

( xm ) n = x m n

Practice:

  1. Simplify \begin{align*}(x^5)^6\end{align*}
  2. Simplify \begin{align*}(x^7)^3\end{align*}
  3. 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.

License: CC BY-NC 3.0

Practice:

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

Exponent Rule #5

License: CC BY-NC 3.0

Practice:

  1. \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.

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