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# 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.

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