Simplifying Radicals with Variables

The video tutorial covers simplifying radicals, including those with variables, using factor trees and grouping. It introduces an alternative method using division and remainders. The tutorial concludes with real-world applications of exponent and radical properties, such as calculating the surface area of a dog's body.

What is the first step in simplifying the radical expression \( \sqrt[3]{54x^9y^5w^7} \) using the factor tree method?

• Combine all the bases into one term.

• Directly apply the cube root to each base.

• Separate the bases inside the radical.

• Ignore the bases and focus on the coefficients.

When simplifying \( x^9 \) under a cube root, how many groups of three x's can be formed?

• One group

• Two groups

• Three groups

• Four groups

In the expression \( \sqrt[3]{54x^9y^5w^7} \), what is the simplified form of \( y^5 \) under the cube root?

• y

• y^2

• y^4

• y^3

What is the simplified form of \( w^7 \) under the cube root?

• w^4

• w^3

• w^2

• w

When simplifying \( \sqrt[3]{48a^5b^7c^3} \), what is the simplified form of \( a^5 \) under the cube root?

• a^4

• a^3

• a

• a^2

In the expression \( \sqrt[4]{243w^5x^{12}y^2z^9} \), what is the simplified form of \( x^{12} \) under the fourth root?

• x^4

• x^3

• x^2

• x

Using the division method, what is the simplified form of \( x^9 \) under the cube root?

• x^4

• x^3

• x^2

• x

When using the division method, what is the remainder when dividing 7 by 3 for the term \( w^7 \)?

• 2

• 3

• 1

• 0

In the surface area problem, what is the first step to convert the mass from kilograms to grams?

• Multiply by 100

• Multiply by 1000

• Divide by 100

• Divide by 1000

What is the approximate surface area of a dog with a mass of 9 kilograms, given the formula \( S = 8.5M^{2/3} \)?

• 865.3 square meters

• 8653 square meters

• 8653 square centimeters

• 865.3 square centimeters