Lesson 3 Radical Thinking Develop Understanding

Jump Start

If $n$ is a positive integer greater than 1 and both $a$ and $b$ are positive real numbers then,

$\sqrt[n]{a^{n}} = a$
$\sqrt[n]{a b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$
$\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$

a.

Illustrate each of these rules for radicals with an example using numbers.

b.

Show why each of these rules is true by rewriting the radical expressions using rational exponents.

Learning Focus

Change the form of radical expressions using properties of exponents.

Which is a more efficient way to change the form of a radical expression: using radicals or using exponents?

How do the properties of exponents help explain methods for representing and manipulating radicals?

Open Up the Math: Launch, Explore, Discuss

Now that Tia and Tehani know that $a^{\frac{m}{n}} = {(\sqrt[n]{a})}^{m}$ they are wondering which form, radical form or exponential form, is best to use when working with numeric and algebraic radical expressions.

Tia says she prefers radicals since she understands the following properties for radicals (and there are not too many properties to remember):

If $n$ is a positive integer greater than $1$ and both $a$ and $b$ are positive real numbers then,

$\sqrt[n]{a^{n}} = a$
$\sqrt[n]{a b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$
$\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$

Tehani says she prefers exponents since she understands the following properties for exponents (and there are more properties to work with):

$a^{m} \cdot a^{n} = a^{m + n}$
${(a^{m})}^{n} = a^{m n}$
${(a b)}^{n} = a^{n} b^{n}$
${(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}}$
$\frac{a^{m}}{a^{n}} = a^{m - n}, a \neq 0$
$a^{- n} = \frac{1}{a^{n}}$

Both Tia and Tehani agree that it is important to understand the following rule that expresses the relationship between radicals and exponents:

$a^{\frac{1}{n}} = \sqrt[n]{a}$ and $a^{\frac{m}{n}} = {(\sqrt[n]{a})}^{m}$

Using their preferred notation, Tia might rewrite $\sqrt[3]{x^{8}}$ as follows:

$\sqrt[3]{x^{8}} = \sqrt[3]{x^{3} \cdot x^{3} \cdot x^{2}} = \sqrt[3]{x^{3}} \cdot \sqrt[3]{x^{3}} \cdot \sqrt[3]{x^{2}} = x \cdot x \cdot \sqrt[3]{x^{2}} = x^{2} \sqrt[\cdot 3]{x^{2}}$

(Tehani points out that Tia also used some exponent rules in her work.)

On the other hand, Tehani might rewrite $\sqrt[3]{x^{8}}$ as follows:

$\sqrt[3]{x^{8}} = x^{\frac{8}{3}} = x^{\frac{2 + 2}{3}} = x^{2} \cdot x^{\frac{2}{3}}$ or $x^{2} \sqrt[\cdot 3]{x^{2}}$

For each of the following problems, change the form of the expression in the ways you think Tia and Tehani might do it.