# Section 1.4: Exponents

The Laws of Exponents let you rewrite algebraic expressions that involve exponents. The last three listed here are really definitions rather than rules.

#### Laws of Exponents

All variables here represent real numbers and all variables in denominators are nonzero.

- \(x^a\cdot x^b=x^{a+b}\)
- \(\dfrac{x^a}{x^b}=x^{a-b}\)
- \(\left(x^a\right)^b=x^{ab}\)
- \((xy)^a=x^a y^a\)
- \(\left(\dfrac{x}{y}\right)^b=\dfrac{x^b}{y^b}\)
- \(x^0=1\), provided \(x\neq 0\). [Although in some contexts \(0^0\) is still defined to be 1.]
- \(x^{-n}=\dfrac{1}{x^n}\), provided \(x\neq 0\).
- \(x^{1/n}=\sqrt[n]{x}\), provided \(x\neq 0\).

#### Example 1

Simplify \(\left(2x^2\right)^3(4x)\).

We'll begin by simplifying the \(\left(2x^2\right)^3\) portion. Using Property 4, we can write

\(2^3\left(x^2\right)^3(4x)\) | |

\(8x^6(4x)\) | Evaluate \(2^3\), and use Property 3. |

\(32x^7\) | Multiply the constants, and use Property 1, recalling \(x = x^1\). |

Being able to work with negative and fractional exponents will be very important later in this course.

#### Example 2

Rewrite \(\dfrac{5}{x^3}\) using negative exponents.

Since \(x^{-n}=\dfrac{1}{x^n}\), then \(x^{-3}=\dfrac{1}{x^3}\) and thus \[\dfrac{5}{x^3}=5x^{-3}.\]

#### Example 3

Simplify \(\left(\dfrac{x^{-2}}{y^{-3}}\right)^2\) as much as possible and write your answer using only positive exponents. \begin{align*} \left(\dfrac{x^{-2}}{y^{-3}}\right)^2=& \dfrac{\left(x^{-2}\right)^2}{\left(y^{-3}\right)^2}\\ =& \dfrac{x^{-4}}{y^{-6}}\\ =& \dfrac{y^6}{x^4} \end{align*}

#### Example 4

Rewrite \(4\sqrt{x}-\dfrac{3}{\sqrt{x}}\) using exponents.

A square root is a radical with index of two. In other words, \(\sqrt{x}=\sqrt[2]{x}\). Using the exponent rule above, \(\sqrt{x}=\sqrt[2]{x}=x^{1/2}\). Rewriting the square roots using the fractional exponent, \[4\sqrt{x}-\dfrac{3}{\sqrt{x}}=4x^{1/2}-\dfrac{3}{x^{1/2}}.\]

Now we can use the negative exponent rule to rewrite the second term in the expression: \[4x^{1/2}-\dfrac{3}{x^{1/2}}=4x^{1/2}-3x^{-1/2}.\]

#### Example 5

Rewrite \( \left(\sqrt{p^5}\right)^{-1/3} \) using only positive exponents.

\begin{align*} \left(\sqrt{p^5}\right)^{-1/3}=& \left(\left(p^5\right)^{1/2}\right)^{-1/3}\\ =& p^{-5/6}\\ =& \frac{1}{ p^{5/6}} \end{align*}

#### Example 6

Rewrite \( x^{-4/3} \)as a radical.

\begin{align*} x^{-4/3}=& \frac{1}{x^{4/3}} \\ =& \frac{1}{\left(x^{1/3}\right)^4} \quad \text{(since \(\frac{4}{3}=4\cdot\frac{1}{3}\))}\\ =& \frac{1}{\left(\sqrt[3]{x}\right)^4} \quad \text{(using the radical equivalence)} \end{align*}