Simplifying Roots and Rationalizing Denominators

The video tutorial explains how to rationalize denominators by converting expressions into forms with perfect cubes or fourth roots. It covers the process of prime factorization to identify necessary multipliers for both the numerator and denominator, ensuring the radicand becomes a perfect power. Two examples are provided: one with cube roots and another with fourth roots, demonstrating the step-by-step simplification to achieve a rationalized denominator.

What is the first step in rationalizing the denominator of the cube root of 5 divided by the cube root of 28?

• Add 5 to both numerator and denominator

• Express the denominator as a perfect cube

• Write the expression as a single cube root

• Multiply both numerator and denominator by 28

What is the prime factorization of 28?

• 5 x 5 x 5

• 4 x 4 x 4

• 3 x 3 x 3

• 2 x 2 x 7

To make the denominator a perfect cube, what should be multiplied with the cube root of 28?

• Cube root of 2 x 2 x 2 x 7 x 7

• Cube root of 2 x 7 x 7

• Cube root of 2 x 2 x 7

• Cube root of 7 x 7 x 7

What is the simplified form of the cube root of 490 divided by 14?

• Cube root of 490 divided by 14

• Cube root of 490 divided by 7

• Cube root of 245 divided by 14

• Cube root of 245 divided by 7

Why does the cube root of 490 not simplify further?

• It is already a perfect cube

• It has no prime factors

• It is a prime number

• It does not have three of any factors

What is the first step in rationalizing the denominator of the fourth root of 5 divided by the fourth root of 12?

• Express the denominator as a perfect fourth root

• Add 5 to both numerator and denominator

• Multiply both numerator and denominator by 12

• Write the expression as a single fourth root

What is the prime factorization of 12?

• 2 x 2 x 3

• 5 x 5 x 5

• 3 x 3 x 3

• 4 x 4 x 4

To make the denominator a perfect fourth root, what should be multiplied with the fourth root of 12?

• Fourth root of 2 x 2 x 2 x 2 x 3 x 3 x 3

• Fourth root of 2 x 2 x 3

• Fourth root of 2 x 2 x 2 x 3 x 3

• Fourth root of 3 x 3 x 3 x 3

What is the simplified form of the fourth root of 540 divided by 6?

• Fourth root of 540 divided by 6

• Fourth root of 540 divided by 3

• Fourth root of 270 divided by 6

• Fourth root of 270 divided by 3

Why does the fourth root of 540 not simplify further?

• It is already a perfect fourth root

• It has no prime factors

• It does not have four of any factors

• It is a prime number