Chapter 2.2

Objectives

Review the lesson material
Defeat the Area challenge
Complete the lesson reflection
Submit the assignment

Key Vocabulary

Rational Exponent

An exponent with a fraction, where the numerator represents a power, and the denominator represents a root.
Radical

A way to write the root of a number. Eg, $LaTeX: \sqrt{25}$ means the square root of 25.
Exponent Properties

Rules for how to work with exponents, such as the product rule, quotient rule, and power rule.

Lesson Outcomes & Standards

Students will be able to express radicals in terms of rational exponents
Students will Define rational exponents and explain their relationship to radicals $LaTeX: 5^{\frac{1}{3}}=\sqrt[3]{5}$
Apply the properties of exponents to simplify expressions involving rational exponents (eg. $LaTeX: \left(5^{\frac{1}{3}}\right)^3=5^{\frac{3}{3}}=5$

Common Core:

N.RN.1 - Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^1/3 to be the cube root of 5 because we want (5^1/3)^3 = 5(^1/3)^3 to hold, so (5^1/3)^3 must equal 5.

The story continues...

You did a great job replanting on this side of the river, but you will need to figure out a way across if you are going to finish replanting. The Fraction Faction appears to have destroyed the bridge, so you will have to find another way across.

How will you get to the other side?

From Dream to Reality

A kid pitches a skate park to the mayor. — **Figure 1.** *By including an estimated cost in their proposals, students can better convenience politicians.*

Every year, high school students passionate about skateboarding pitch their local governments for a safe place to practice. Most of the time, students just ask for a skate park, but every so often, a student takes the time to craft a proposal that is so in depth and detailed that the city can easily make a decision and get a skate park funded.

One of the biggest hurdles to funding is the cost of concrete. By showing concrete cost calculations, students create plans that optimize cost-effectiveness while ensuring the park is safe and durable. By extending the properties of integer exponents to rational exponents, one student calculated the perfect amount of concrete, the angles of ramps, and the area required for smooth landings. The city officials were impressed not only by his passion but also by his precise calculations and detailed plan. With the funds secured, they transformed an empty lot into a new skateboard park. Without an understanding of roots, powers, and volume, the city's first skate park may never have been made.

Why use roots?

Square and Cube roots were handy ways to address our need to find the value of a number when we know it has been squared or cubed. It simplifies the solving of equations involving areas and volumes. Now that we have learned about exponent rules, we can do more advanced and useful calculations that would have been cumbersome using the radical $LaTeX: \sqrt{ }$ symbol. Radicals are the opposite operation of exponents.

Roots are just exponents with fractions

A square root is the same as a base to the power of 1/2. In other words:

$LaTeX: \sqrt{a}=a^{\frac{1}{2}}$

$LaTeX: \sqrt[3]{a}=a^{\frac{1}{3}}$

$LaTeX: \sqrt[5]{a}=a^{\frac{1}{5}}$

You may have noticed that with the square root, there was no index on the radical sign. When it's not written, it's assumed to be an index of two or a square root.

We can take this a step further using the exponent rules we learned in a previous lesson.

$LaTeX: \sqrt[3]{a^2}=\left(a^2\right)^{^{\frac{1}{3}}}=a^{\frac{2}{3}}$

Or go even more advanced and apply several rules

$LaTeX: \sqrt[5]{a^{10}b^{-2}}=\left(\frac{a^{10}}{b^2}\right)^{^{\frac{1}{5}}}=\frac{a^{\frac{10}{5}}}{b^{\frac{2}{5}}}=\frac{a^2}{b^{\frac{2}{5}}}$

We will cover more about exponent rules in the next lesson. Be sure to practice them until you have each rule memorized!

Master Essential Skills

Question 1: Concrete Estimations

Alex is a general contractor and was just hired to give a quote on a concrete patio. The drawings show that the patio will be in the shape of a square, and the area of the patio will be 49 square feet. Alex needs to know if he should quote 8ft, 10ft, or 12ft board. How long will each side be?

Question 2: A Birthday Surprise

Emma is planning a surprise party and wants to fill a large spherical balloon with helium and confetti. She knows that the balloon needs to have a volume of 24,000 cubic inches to be large enough for the decoration she has in mind. However, the packaging does not provide the radius of the balloon. Emma needs to calculate the radius to ensure it will fit through her front door, which is 36 inches wide.

Essential information:

The volume of a sphere is given by $LaTeX: V=\frac{4}{3}\pi r^3$

Check Your Work

Question 1. Calculate the length of the patio

Since it's a square, each side will be equal. You can take the square root.

$LaTeX: length=\sqrt{49ft^2}=\:7ft$

Therefore he needs to buy 8ft boards to avoid as much waste as possible.

Question 2: Find the radius

Step 1: Rearrange for r.

$LaTeX: V=\frac{4}{3}\pi r^3$ → $LaTeX: \frac{\color{red}{3}}{\color{blue}{4}\color{orange}{\pi}} \times V = \frac{\cancel{4}}{\cancel{3}} \cancel{\pi} r^3 \times \frac{\cancel{\color{red}{3}}}{\cancel{\color{blue}{4}}\cancel{\color{orange}{\pi}}}$

$LaTeX: r^3=\frac{3V}{4\pi}$

Step 2: Add in your volume

$LaTeX: r^3=\frac{3\times24000}{4\pi}\approx5729.6in^3$

Step 3: Take the cube root

$LaTeX: r=\sqrt[3]{5729.6}$ or use exponents $LaTeX: r=5729.6^{\frac{1}{3}}$

r = 17.9 inches

This means the diameter is twice that. So the balloon will need 35.8 inches of clearance.

It'll be a tight fit, but the balloon will clear the door.

Did you know that...

Did you know that the concept of square roots dates back to ancient civilizations? The Babylonians developed methods for approximating square roots as early as 2000 BC!

Practice

Fractional exponent to radical expression

Show how the following fractional exponent can be written as its equivalent radical expression:

$LaTeX: \left(3^{^{\frac{1}{5}}}\right)^{^2}=\sqrt[5]{9}$

Solution:

Step 1

The fractional exponent can be rewritten as follows using the power of a power rule:

$LaTeX: \left(3^{^{\frac{1}{5}}}\right)^2=3^{^{\frac{1}{5}\cdot2}}$

Step 2

Using the same power of power rule, this can further be rewritten as follows:

$LaTeX: 3^{^{\frac{1}{5}\cdot2}}=\left(3^2\right)^{^{\frac{1}{5}}}=9^{^{\frac{1}{5}}}$

Step 3

A fractional exponent can be rewritten as a radical with the denominator as the index.

i.e.,

$LaTeX: 9^{^{\frac{1}{5}}}=\sqrt[5]{9}$

Step 4

To verify the above step, take the 5th root of 9.

$LaTeX: \left(9^{^{\frac{1}{5}}}\right)^{^5}=9^{^{\frac{1}{5}\cdot5}}=9$

$LaTeX: \therefore9^{^{\frac{1}{5}}}is\:the\:5th\:root\:of\:9$

Conclusion

The above steps explain this equivalency:

$LaTeX: \left(3^{^{\frac{1}{5}}}\right)^{^2}=\sqrt[5]{9}$

Simplify radical expressions with variables

Rewrite the following radical expression in rational exponent form and simplify. Assume that m is positive. Let n=4m.

$LaTeX: \sqrt[3n]{a^{2m}}$

Solution:

Step 1

Rewrite the radical expression with n=4m:

$LaTeX: \sqrt[3n]{a^{2m}}=\sqrt[3\left(4m\right)]{a^{2m}}$

Step 2

Simplify further:

$LaTeX: \sqrt[3\left(4m\right)]{a^{2m}}=\sqrt[12m]{a^{2m}}$

Step 3

A radical expression can be rewritten as a rational exponent with the index as the denominator of the rational exponent, and the exponent of the radical as the numerator.

The index is 12m.

The exponent of the radical is 2m.

$LaTeX: \sqrt[12m]{a^{2m}}=a^{^{\frac{2m}{12m}}}$

Step 4

Simplify further:

$LaTeX: a^{^{\frac{2m}{12m}}}=a^{^{\frac{1}{6}}}$

Conclusion

Therefore the simplified equivalent rational exponent form is:

$LaTeX: \sqrt[3n]{a^{2m}}=a^{^{\frac{1}{6}}}$

Test a claim

A math novice wrote the following:

$LaTeX: \sqrt[4]{x^2}=x^{^{\frac{2}{4}}}=x^{^{\frac{1}{2}}}=\sqrt{x}$

Based on this, the novice claims that,

$LaTeX: \sqrt[4]{x^2}=\sqrt{x}for\:all\:values\:of\:x.$

Give an example of a value of x that makes this statement untrue and explain your reasoning.

Solution:

Step 1

Test the claim for a positive value of x.

$LaTeX: \sqrt[4]{x^2}=\sqrt{x}$

Step 2

Consider x=3;

$LaTeX: \sqrt[4]{3^2}=\sqrt{3}$

The above can also be written as:

$LaTeX: \left(\sqrt[4]{3}\right)^{^2}=\sqrt{3}$

Step 3

Rewriting the left side of the above equation in rational expression form:

$LaTeX: \left(3^{^{\frac{1}{4}}}\right)^{^{^2}}=\sqrt{3}$

$LaTeX: 3^{^{\frac{1}{4}\cdot2}}=\sqrt{3}$

$LaTeX: 3^{^{\frac{1}{2}}}=\sqrt{3}$

$LaTeX: \sqrt{3}=\sqrt{3}$

Therefore, the claim holds true for positive numbers.

Step 4

Test the claim for a negative value of x.

Consider x=-9

$LaTeX: \sqrt[4]{\left(-9\right)^2}=\sqrt{-9}$

The above can also be written as:

$LaTeX: \sqrt[4]{81}=\sqrt{-9}$

Step 5

Simplifying further:

$LaTeX: 3\ne\sqrt{-9}$

This shows that the statement is untrue for negative numbers.

Conclusion

The above claim is untrue for negative values of x.

Radical Root Rapsody Remix

Read the Lyrics

Verse 1:

Numbers high in the sky, that’s what exponents do,
But when it’s a fraction, I’ve got a trick for you.
Numerator’s the power, it’s what makes the base climb,
Denominator’s the root, dividing it every time!

Pre-Chorus: (Building understanding)

Take five to the one-third, it’s just like you heard,
The cube root of five, math magic, that’s the word!
Numerator, lift it high, denominator keeps it low,
Now you’ve got the rules, watch how your skills grow!

Chorus:

Rational exponents, breaking them down,
Fractions and roots, turning it around.
Power on the top, root on the ground,
Learn it with the song, and you’ll be math-bound!

Verse 2:

Square root? That’s easy, just a one-half in disguise,
Cube root? Use one-third, and watch the number rise.
Take the power and the root, multiply them just right,
Rational exponents, make your math shine bright!

Bridge: (Interactive, with call-and-response)

Let’s test it out—what’s five to the one-half?
(Students: Square root of five!)
What about eight to the one-third?
(Students: Cube root of eight!)
You’ve got it now, see how it’s done,
Rational exponents, learning’s so much fun!

Chorus:

Rational exponents, breaking them down,
Fractions and roots, turning it around.
Power on the top, root on the ground,
Learn it with the song, and you’ll be math-bound!

Outro: (Slow it down for emphasis)

Now you know the secret, now you’ve got the key,
Rational exponents, it’s as easy as can be.
Math is cool, and now you know,
With this song in your heart, watch your knowledge grow!

Reflection Questions

When might you wish to use the quadratic formula verses another strategy (such as completing the square or factoring)?
What strategies might you use to check your work when using the quadratic formula?
Can you think of any real-world situations where the quadratic formula might be used?
How do you decide which of the two solutions to use in a given situation? Are there situations when you may need both?

What's next?

Complete the practice questions

Continue to the practice section

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