Simplifying Radicals and Prime Factorization

The video tutorial explains how to simplify radicals containing variables. It covers the general rule that if the index is even and a variable outside the radical has an odd exponent, an absolute value is needed. Two examples are provided: simplifying the fourth root of 81x^8y^4, which requires an absolute value, and the fifth root of 64x^10y^6, which does not. The process involves prime factorization and grouping factors according to the index.

What is a key consideration when simplifying radicals with variables?

• Never use absolute values.

• Use absolute values if the index is even and the variable has an odd exponent.

• Use absolute values only if the index is odd.

• Always use absolute values.

What is the prime factorization of 81?

• 3 x 3 x 3

• 9 x 9

• 3 x 3 x 3 x 3

• 4 x 4 x 4

How do you express x^8 for simplifying the fourth root?

• x^4 x^4

• x^3 x^3 x^2

• x^2 x^2 x^2 x^2

• x x x x x x x x

Why is an absolute value needed in the simplified expression of the fourth root example?

• Because the index is even.

• Because x has an odd exponent outside the radical.

• Because y has an odd exponent outside the radical.

• Because the index is odd.

What is the simplified form of the fourth root of 81x^8y^4?

• 3x^2y

• 3x^2|y|

• 3|x^2||y|

• 3|x^2|y

What is the prime factorization of 64?

• 4 x 4 x 4

• 8 x 8

• 16 x 4

• 2 x 2 x 2 x 2 x 2 x 2

How do you express x^10 for simplifying the fifth root?

• x^3 x^3 x^4

• x^2 x^2 x^2 x^2 x^2 x^2

• x^5 x^5

• x^2 x^2 x^2 x^2 x^2

Why is no absolute value needed in the simplified expression of the fifth root example?

• Because the index is even.

• Because the index is odd.

• Because x has an even exponent outside the radical.

• Because y has an even exponent outside the radical.

What is the simplified form of the fifth root of 64x^10y^6?

• 2x^2y

• 2|x^2||y|

• 2x^2|y|

• 2|x^2|y

Which of the following is true about the principal root?

• It is always negative.

• It is always zero.

• It is always positive.

• It can be either positive or negative.