Lesson 7 Radical Ideas Practice 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?

Technology guidance for today’s lesson:

Convert Rational Exponents to Radicals: Casio ClassPad Casio fx-9750GIII

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:

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

1.

Original Expression:

$\sqrt{27}$

What Tia and Tehani might do to change the form of the expression:

Tia’s method:

Tehani’s method:

2.

Original Expression:

$\sqrt[3]{32}$

What Tia and Tehani might do to change the form of the expression:

Tia’s method:

Tehani’s method:

3.

Original Expression:

$\sqrt{20 x^{7}}$

What Tia and Tehani might do to change the form of the expression:

Tia’s method:

Tehani’s method:

4.

Original Expression:

$\sqrt[3]{\frac{16 x y^{5}}{x^{7} y^{2}}}$

What Tia and Tehani might do to change the form of the expression:

Tia’s method:

Tehani’s method:

Pause and Reflect

One of the conventions in the math world is to rewrite expressions so they do not have negative exponents. One of the easiest ways to get started is to rewrite the negative exponents as positive in the first step, but you can work with each expression in the way you choose. In each step, write a justification for the step. Here, an example of one is entirely worked out.

	Problem:	Step 1:	Step 2:	Answer:
EXAMPLE	$\frac{x}{x^{- 3}}$	Rewrite with positive exponents. $x \cdot x^{3}$ Because $x^{3} = \frac{1}{x^{- 3}}$	Apply the exponent rules. $x^{1 + 3}$ Exponent multiplication rule	$x^{4}$

Problem:

Step 1:

Step 2:

Answer:

EXAMPLE

$\frac{x}{x^{- 3}}$

Rewrite with positive exponents.

$x \cdot x^{3}$

Because $x^{3} = \frac{1}{x^{- 3}}$

Apply the exponent rules.

$x^{1 + 3}$

Exponent multiplication rule

$x^{4}$

5.

Problem:	Step 1:	Step 2:	Answer:
$\frac{y^{- 3}}{y^{- 1}}$	Rewrite with positive exponents.	Apply the exponent rules.	$\frac{1}{y^{2}}$

6.

Problem:	Step 1:	Step 2:	Answer:
$\frac{3 x^{- 9}}{6 x^{- 11}}$			$\frac{x^{2}}{2}$

7.

Problem:	Step 1:	Step 2:	Answer:
$\frac{a^{2} b^{- 3}}{a^{3} b^{- 1}}$

8.

Problem:	Steps			Answer:
${(\frac{2 x}{y})}^{- 5}$	Raise everything to the $- 5$ power.	Rewrite with positive exponents.	Combine common factors.

9.

Problem:	Steps			Answer:
${(\frac{c^{4} d}{c^{- 2}})}^{- 4}$

Ready for More?

Now we have a challenge: The answer to a problem is given in the last column. Your job is to create “the problem.” The problem must be equivalent to the answer given and contain negative exponents. After writing the problem, show how to change the form to get the answer provided. Challenge yourself to come up with the trickiest problems that you can!

Problem:	Steps:			Answer:
				$\frac{3 d^{7}}{25 c^{6}}$
				$\frac{x^{2} z^{7}}{y^{5}}$
				$1$

Takeaways

Strategies for rewriting radical expressions, root $n$ , in equivalent forms:

Strategies for changing the form of exponential expressions:

Vocabulary

radical
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned how to change the form of complicated radical and exponential expressions using the properties of radicals and exponents. Strategies for changing the form of radical expressions can be explained by converting the radical expressions to exponential form.

Retrieval

1.

Fill in the tables based on the given explicit equation. Be prepared to explain how the numbers in the equations connect with the values in the tables.

a.

$f (x) = - 5 x + 17$

$x$	$f (x)$
$- 1$
$0$
$1$
$2$
$3$

b.

$f (x) = - 5 (x - 2) + 17$

$x$	$f (x)$
$- 1$
$0$
$1$
$2$
$3$

c.

$f (x) = - 5 (x + 1) + 17$

$x$	$f (x)$
$- 1$
$0$
$1$
$2$
$3$

2.

Find the $x$ - and $y$ -intercepts and then graph the function.

a.

$f (x) = 2 (x - 3)$

$x$ -intercept:

$y$ -intercept:

b.

$f (x) = - 3 (x + 1) + 6$

$x$ -intercept:

$y$ -intercept: