Properties of Radicals

RADICAL EXPRESSIONS

What are radical expressions, their properties, and how do you simplify them?

Open Ended

Based on your own understanding, what is a radical expression?

What is a radical expression?

• A radical is a symbol that represents the root of a number

• A radical expression has three important components: index (root), radical, radicand

Multiple Choice

Examine the radical expression. What is the index?

• 5

• 108

• 36

Multiple Choice

Examine the radical expression. What is the radicand?

• 3

• 108

• 36

What does a radical expression mean?

A radical expression (or root) of a number is the value that can be raised to the index (root) as an exponent to obtain the value in the radicand

Multiple Choice

Examine the radical expression. Which question best represents this expression?

• What value can be multiplied by itself 108 times to obtain 5?

• What value can be multiplied by itself 5 times to obtain 108?

• What is the value of 108 divided by 5?

• What is the value of 5 divided by 108?

How are exponents and radicals related to each other?

Remember that exponents are the number of times that a number is multiplied by itself.

Radicals undo exponents, representing what number can be multiplied by itself to obtain another value.

Converting Between Exponential and Radical Expressions

Multiple Choice

Write the given expression in exponential form.

• $$8^{\frac{10}{3}}$$
  

• $$10^{\frac{3}{8}}$$

• $$10^{\frac{8}{3}}$$
  

• $$3^{\frac{8}{10}}$$

Multiple Choice

Write the given expression in exponential form.

• $$h^{\frac{1}{2}}$$

• $$h^2$$

• $$h$$

• $$h^{\frac{1}{3}}$$

Multiple Choice

Write the given expression in radical form.

Multiple Choice

Write the given expression in radical form.

Properties of Radicals

Inverse Rule, Product Rule, Quotient Rule, Power Rule

Inverse Property

The nth root of a value raised to the nth power is equivalent to the value itself

Multiple Choice

Which property of exponents could be used to prove the inverse property of radicals?

• Product of Powers

• Quotient of Powers

• Power of Power

Multiple Choice

Simplify the radical expression.

• $$18$$

• $$18^5$$

• $$\sqrt{18}$$

• $$3$$

Multiple Choice

Simplify the radical expression.

• $$6c$$

• $$6c^2$$

• $$36c^2$$

• $$\sqrt{6c}$$

Multiple Choice

Simplify the radical expression.

• $$d^3r$$  
  

• $$d^{12}r^4$$  
  

• $$\sqrt{d^3r}$$  
  

• $$\sqrt{d^{12}r^4}$$  
  

Radical of Product Property

The nth root of a product is equivalent to the product of the nth roots

Multiple Choice

Which property of exponents could be used to prove the radical of product property of radicals?

• Power of Product

• Power of Quotient

• Power of Power

Multiple Choice

Rewrite (expand) the radical expression using the Radical of Product property.

Multiple Choice

Rewrite (condense) the radical expression using the Radical of Product property.

Multiple Choice

Which of the following is NOT a possible way to rewrite the given expression?

Multiple Choice

Which of the following is NOT a possible way to rewrite the given expression?

Radical of Quotient Property

The nth root of a quotient is equivalent to the quotient of the nth roots

Multiple Choice

Which property of exponents could be used to prove the radical of quotient property of radicals?

• Power of Product

• Power of Quotient

• Power of Power

Multiple Choice

Which of the following is NOT a possible way to rewrite the radical expression?

Multiple Choice

Rewrite (expand) the radical expression using the Radical of Quotient property.

Multiple Choice

Rewrite (condense) the radical expression using the Radical of Quotient property.

Radical of Power Property

The nth root of a power is equivalent to the power of the nth root

Multiple Choice

Which property of exponents could be used to prove the radical of power property of radicals?

• Power of Product

• Power of Quotient

• Power of Power