Primary Exponent Rules (Product, Quotient, and Power of a Power)


In this section, we will look at simplifying algebraic expressions using the product, quotient, and power exponent rules.

Power, Quotient, and Power Exponent Rules 

For any real numbers \(a\), \(b\), \(c\), \(x\), and \(y\), where \(x\neq0\), \(y\ne0\): 

  1. Product Rule for Exponents
    \((x^a)(x^b)=x^{a+b}\)
  2. Quotient Rule for Exponents
    \(\dfrac{x^a}{x^b}=x^{a-b}\)
  3. Power of a Power Rule for Exponents
    \((x^a)^b=x^{ab}\)
  4. Power of a Product Rule For Exponents
    \((x^ay^b)^c=x^{ac}y^{bc}\)
  5. Power of a Quotient Rule for Exponents
    \(\left (\dfrac{x^a}{y^b}\right)^c=\dfrac{x^{ac}}{y^{bc}}\)

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Example 1

Simplify \((6x^2y^3)(2x^5y)\).

 

Example 2

Simplify \(\dfrac{27a^5b^7}{81a^4b^2}\).

 

 

Example 3

Simplify \(\dfrac{(x^2y)^5}{(y^2x^3)^2}\) first, then evaluate for \(x=-2\) and \(y=3\).

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Example 4

Simplify \((2m^2p^4m)^3\).

Solution

There are powers with the same base \(m\) inside of the brackets which need to be simplified first.

\[\begin{align*} (2m^2p^4m)^3&=(2m^{2+1}p^4)^3 && \text{product rule}\\ &=(2m^3p^4)^3\\ &=2^{(1)(3)}m^{(3)(3)}p^{(4)(3)} && \text{power of a product rule}\\ &=2^3m^9p^{12} \\ &=8m^9p^{12} && \text{evaluate the constant term} \end{align*}\]

Example 5

Simplify \(\dfrac{(5a^2b^3)(2a^3b^2)}{(4ab)^2}\).

Solution

Begin by simplifying powers with the same base in the numerator and applying the exponent to the term inside the brackets in the denominator.

\[\begin{align*} \frac{(5a^2b^3)(2a^3b^2)}{(4ab)^2}&=\frac{(5\times 2)a^{2+3}b^{3+2}}{4^{(1)(2)}a^{(1)(2)}b^{(1)(2)}}\\ &=\frac{10a^5b^5}{4^2a^2b^2}\\ &=\frac{10a^5b^5}{16a^2b^2}\\ &=\frac{10}{16}a^{5-2}b^{5-2} && \text{divide coefficients and use quotient rule}\\ &=\frac{5}{8}a^3b^3&& \text{reduce fraction}\\ &=\frac{5a^3b^3}{8} \end{align*}\]

Check Your Understanding 1


Simplify the following expression completely.

\(\dfrac{(2cd^2)^3}{(2cd^3)(4d)}\)

Enter \(a^b\) as "\(a^{\wedge} b\)".

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