Alg2 L 5.1: Roots, Radicals, and Rational Exponents

Lesson 5.1: Roots, Radicals, and Rational Exponents

Multiple Choice

Simplify the following radical: $$\sqrt[]{300}$$

• 
  $$3\sqrt[]{10}$$

• 
  $$2\sqrt[]{150}$$

• 
  $$100\sqrt[]{3}$$

• 
  $$10\sqrt[]{3}$$

Multiple Choice

$$\sqrt[]{64}$$

• 4
• 6
• 9
• 8

What is a root?

Multiple Choice

The √20 is between which two integers?

• 2 & 3

• 3 & 4

• 4 & 5

• 5 & 6

Multiple Choice

All the following are Perfect Squares except for which one?

• 81
• 121
• 361
• 198

Multiple Choice

-√400

• 200

• -200

• -20

• 20

• not real

Labelling

Can you name the parts of a radical expression?Drag the label to the correct part of the image.

Drag labels to their correct position on the image

radicand

radical symbol

index

Multiple Choice

Select the square root that would be represent point A.

• √36

• √300

• √62

• √115

Multiple Choice

Simplify:

• 8x⁹

• 8x³

• 8x⁴ $$\sqrt[]{x}$$

• 8x³ $$\sqrt[]{x}$$

Multiple Choice

What is the cube root of 8?

• 4
• 2
• -2
• -4

• Used to find roots

• A number/variable in front of the radical symbol is a coefficient, and thus multiplied to the radical expression

• No index? it is assumed to be 2.

Radical Expressions

Multiple Choice

What is the cube root of 64?

• 2
• 4
• 8
• 16

Multiple Choice

$$\sqrt{45x^4y^7}$$  

• $$3x²y³\sqrt{5}$$  

• $$9x²y³\sqrt{5y}$$  

• $$3x²y³\sqrt{5y}$$  

• $$3x⁴y⁷$$  

Multiple Choice

Which expression with a fractional exponent best represents √7 ?

• 7 ^½
• 7¼
• 7⅓
• 2⁷

A radical asks...

Multiple Choice

What is the value of 27^⅓?

• 27
• 9
• 3
• 1/3

Fill in the Blank

Determine the square root of $$-4\sqrt{36}$$  

Multiple Choice

8^1/3 = ?

• 2.666

• 2

• 8.333

To answer those questions...

Knowing the first few perfect squares and cubes can be useful.

Let's list the first 10 perfect squares and perfect cubes.

Multiple Choice

4^3/2 = ?

• 6

• 8

• 5.5

Explanation Slide...

Remember to do the square root before you do the multplication! #PEMDAS

Multiple Choice

8^2/3 = ?

• 26 / 3

• 4

• 16 / 3

Armed with the knowledge of

what a radical expression is and the list of perfect squares and perfect cubes, let's do a few examples.

Multiple Choice

Simplify the following expression:

• 16
• 32
• 40
• 64

Fill in the Blank

What is the cube root: $$\sqrt[3]{-64}$$  

Multiple Choice

Rewrite in radical form.

• A
• B
• C
• D

Match

Match each of the following radical expressions to its equivalent expression.

• $$-\sqrt[]{121}$$

• $$-\sqrt[]{\frac{4}{25}}$$

• $$\sqrt[3]{343}$$

• $$\sqrt[3]{-125}$$

• $$\sqrt[3]{\frac{1}{27}}$$

• $$-11$$

• $$-\frac{2}{5}$$

• $$7$$

• $$-5$$

• $$\frac{1}{3}$$

Multiple Select

The following expression  $$x^{\frac{5}{2}}$$  is equivalent to:

• $$\frac{5x}{2}$$  

• $$\frac{2x}{5}$$  

• $$\sqrt{x^5}$$  

• $$\ ^5\sqrt{x^2}$$  

Multiple Choice

Find the cube root.

$$\sqrt[3]{\frac{1}{216}}$$  

• 1/4

• 1/6

• 1/7

• 1/8

Multiple Choice

$$x^{\frac{-3}{2}}$$  is equal to:

What did all the previous expressions have in common?

Multiple Choice

Write the expression with exponents

• $$5x^3$$

• $$5x^{\frac{1}{3}}$$

• $$\left(5x\right)^{\frac{1}{3}}$$

• $$\left(5x\right)^3$$

Multiple Choice

•  y $$\sqrt[3]{x^2}$$

•  xy $$\sqrt[3]{y}$$

•  6y $$\sqrt[3]{x}$$

•  y³ $$\sqrt[3]{x^2}$$

Multiple Choice

Write in exponential form
∜(3x)³

• 3x³⁄ ⁴
• (3x)³∕ ⁴
• (3x)⁴⁄ ³
• 3x⁴⁄ ³

What happens when we don't have a perfect square as a radicand?

Remember those perfect squares and cubes?!

• Look for the largest perfect square/cube factor of the radicand.

• Rewrite the expression using 2 radicals.

• Simplify the radical with the perfect square/cube radicand.

• If all radicands have no perfect square/cube factors, they are simplified.

• What do we do with the factors that aren't perfect squares/cubes?

Multiple Choice

Rewrite the following

∛x⁵

• x^3/5

• x^5/3

• x²

• x³⋅x⁵

Multiple Choice

• yz⁶ $$\sqrt[3]{xy^2z^2}$$

• yz⁶ $$\sqrt[3]{xyz}$$

• y²z¹⁰  $$\sqrt[3]{xy^2z^2}$$

• y²z¹⁰  $$\sqrt[3]{xy}$$

Let's do this one together...

Multiple Choice

Cube root of positive number

• Positive

• Negative

• Can't say

And one more...

Multiple Choice

Cube root of negative number

• Positive

• Negative

• Can't say

Multiple Choice

Simplify $$4\sqrt[]{320}$$ .

• $$4\sqrt[]{40}$$

• $$32\sqrt[]{5}$$

• $$12\sqrt[]{5}$$

• $$8\sqrt[]{80}$$

Multiple Choice

$$\sqrt[3]{-250}$$  

• $$5\sqrt[3]{2}$$  

• $$25\sqrt[3]{2}$$  

• $$-5\sqrt[3]{2}$$  

• no solution

Multiple Choice

Simplify $$3\sqrt[3]{108}$$ .

• $$9\sqrt[3]{4}$$

• $$6\sqrt[3]{4}$$

• $$3\sqrt[3]{12}$$

• $$18$$

Multiple Choice

$$\sqrt[4]{x^8y^3z^{13}}$$  

• $$x^4yz^6\sqrt[4]{yz}$$  

• $$x^2z^3\sqrt[4]{y^3z}$$  

• $$x^2yz^3\sqrt[4]{z}$$  

• $$x^2z^3$$  

Multiple Choice

Simplify: $$\sqrt{96}$$  

• $$16\sqrt{6}$$  
  

• $$6\sqrt{16}$$  

• $$9\sqrt{6}$$  

• $$4\sqrt{6}$$  

Multiple Choice

5√48x²

• 10x√3
• 15x√3
• 20x√3
• 25x√3

Multiple Choice

Simplify the following radical: $$\sqrt{32}$$  

• $$3\sqrt{2}$$  
  

• $$4\sqrt{8}$$  
  

• $$16\sqrt{2}$$  
  

• $$4\sqrt{2}$$  
  

Multiple Choice

What is the radicand in the following radical?

$$6\sqrt[]{15}$$  

• 6

• 15

• 2

• 90

Multiple Choice

Simplify:
$$\sqrt{100}$$  

• 1

• 10

• 10²

• Already simplified

Multiple Choice

Write as a radical expression y²^/³

• 2y³
• √y³
• ∛y²
• 3y²

Sometimes radical expression will contain variables:

• If the exponent is divisible by the index, divide.

• If the exponent is not divisible by the index, "break up" the exponent into a value divisible by the index and whatever remains
  

Multiple Choice

∜m³ is the same as which answer?

• m^3/4
• m¹²
• m^4/3
• m⁷

Let's do this one together...

Fill in the Blank

Anything raised to the power of 0 equals ____.

And one more...

Multiple Choice

What is this symbol called: $$\sqrt{4}$$ (ignore the 4) 

• radical

• radicand

• square root

• root

Multiple Choice

Simplify $$\sqrt[]{324a^3b^7}$$ .

• $$18ab^3$$

• $$18ab^3\sqrt[]{ab}$$

• $$18\sqrt[]{a^3b^7}$$

• $$324a^2b^6$$

Explanation Slide...

​

Multiple Choice

Simplify

• $$x^2y\sqrt[]{xy}$$

• xy
• x²y²
• Can't Simplify

Multiple Choice

Given $$\sqrt[c]{a^b}$$  

which of the following is the index

• a

• b

• c

Multiple Choice

• A

• B

• C

• D

Explanation Slide...

​

Multiple Choice

• A

• B

• C

• D

Multiple Choice

$$a^{\frac{b}{c\ }}$$ can be written in radical form as?

• $$\sqrt{a^c}$$  

• $$\sqrt{a^{b\ }}$$  

• $$\sqrt[b]{a^c}$$  

• $$\sqrt[c]{a^b}$$  

Multiple Choice

Find $$b$$ in $$22b^2=35$$ , where $$b$$ is a positive number.

• $$\frac{35}{22}$$

• $$\left(\frac{35}{22}\right)^2$$

• $$\frac{\sqrt[]{35}}{22}$$

• $$\sqrt[]{\frac{22}{35}}$$

• $$\sqrt[]{\frac{35}{22}}$$

Explanation Slide...

​

Multiple Choice

If $$x$$ is a real number, what is $$\sqrt[3]{x^{27}}$$ equivalent to?

• $$x^{-9}$$

• $$x^{\frac{1}{9}}$$

• $$\left|x^9\right|$$

• $$x^9$$

• $$x^{24}$$

Multiple Choice

Given the radical expression $$\sqrt[4]{x^3}$$  ,

what does the 4 represent?

• Index

• Radicand

• Power

• Base

Multiple Choice

If a right triangle has legs of length 5x and x, which of the following expressions represents the length of its hypotenuse in terms of x?

• $$2x$$

• $$5x$$

• $$x\sqrt[]{6}$$

• $$2x\sqrt[]{6}$$

• $$x\sqrt[]{26}$$

Explanation Slide...

​

Now let's talk about Rational Exponents

What does rational mean?
What does exponent mean?

Multiple Choice

$$\left(\sqrt[3]{2}\right)^5$$  

• 2^3/5
• (-2)^3/5

• 2^5/3
• 8⁵

Rational Exponents

Expressions with rational exponents can be rewritten as radicals using the following rules:

​Exponential Form	​Meaning	​Radical Form
​	​	​
​	​	​

Explanation Slide...

​

Easy way to remember...

Multiple Choice

Write $$\sqrt[]{m^5}$$   in exponential form:

• $$m^{\frac{5}{2}}$$  

• $$m^5$$  

• $$m^{\frac{2}{5}}$$  

• $$m^{\frac{1}{5}}$$  

Let's do this one together...

Write the expression in radical form. Simplify if needed.

Explanation Slide...

Reminder: A blank index is really an index number of 2.

And one more...

Write the expression in radical form. Simplify if needed.

Multiple Choice

Simplify the following expression:

• 5
• 25
• 125
• 625

Now let's write in exponential form...

Multiple Choice

Convert $$\sqrt[4]{3x}$$   to exponential form.

• $$3x^{\frac{1}{4}}$$  

• $$3x^4$$  

• $$\left(3x\right)^4$$  

• $$\left(3x\right)^{\frac{1}{4}}$$  

Multiple Choice

Write the expression in radical form, simplifying if necessary:

$$7^{\frac{2}{3}}$$

• $$\sqrt[]{7^3}$$

• $$\sqrt[3]{49}$$

• $$7$$

• $$\sqrt[]{49}$$

Multiple Choice

Convert to radical form: $$3x^{\frac{1}{5}}$$  

• $$3\sqrt[5]{x}$$  

• $$\sqrt[3]{5x}$$  

• $$\sqrt[5]{3x}$$  

• $$\sqrt[]{3x^5}$$  

Multiple Choice

Write with a rational exponent.

• 7³

• 7

• 7^3/4

• 7^4/3

Multiple Choice

First step to Convert to exponential form: $$\sqrt[4]{37x^{21}}$$  

• $$37\left(x^{21}\right)^{\frac{1}{4}}$$  

• $$\left(37x^{21}\right)^{\frac{1}{4}}$$  

• $$\left(37x^{21}\right)^4$$  

• $$\frac{37x^{21}}{4}$$  

Multiple Choice

Write the expression in radical form.

x^7/2

• $$\sqrt[]{x^7}$$

• $$7\sqrt[]{x^2}$$

• $$7x^2$$

• $$\sqrt[3]{x^7}$$

Multiple Choice

Which of the following is in exponential form?

• $$\sqrt[4]{8^{ }}$$  

• $$8^{\frac{1}{4}}$$  

Multiple Choice

Write the following expression in exponential form:

• a.)

• b.)

• c.)

• d.)

Multiple Choice

Which of the following is in radical form?

• $$\sqrt[4]{5^3}$$  

• $$5^{\frac{3}{2}}$$  

Multiple Choice

Write the expression in exponential form.

$$\sqrt[3]{16}$$

• $$4$$

• $$16^{\frac{1}{3}}$$

• $$16^3$$

• $$16$$

Multiple Choice

Write the following expression in radical form: $$81^{\frac{1}{2}}$$  

• $$\sqrt[]{81}$$  

• $$\sqrt[]{81^2}$$  

• $$\sqrt[1]{81^2}$$  

• $$81^2$$  

Multiple Choice

Write with a rational exponent.

HINT: what root is it? It is "invisible"

• 216

• 6^1/2

• 6²

• 6^3/2

Match

Match each expression to it's radical form:

• $$\sqrt[]{25^5}$$  

• $$\sqrt[5]{25^2}$$  

• $$\sqrt[]{5^{25}}$$  

• $$\sqrt[25]{5^2}$$  

 

 

 

 

• $$25^{\frac{5}{2}}$$  

• $$25^{\frac{2}{5}}$$  

• $$5^{\frac{25}{2}}$$  

• $$5^{\frac{2}{25}}$$  

Multiple Choice

$$36^{\frac{1}{2}}\ =$$  

• 6
• $$\sqrt[]{6}$$

• $$18$$

• $$72$$

Multiple Choice

What is $$25^{\frac{1}{2}}-27^{\frac{1}{3}}$$ ?

• -4

• 2

• 8

• $$\frac{7}{6}$$

• $$\frac{52}{3}$$