Simplifying the Cube Root of 500

To find the simplest rationalizing factor of the cube root of 500, we need to identify the prime factors of 500 and then simplify them.

Prime Factorization of 500:
To begin, let's find the prime factors of 500. Prime factors are the prime numbers that, when multiplied together, give the original number.

1. Divide 500 by 2: 500 ÷ 2 = 250
- 2 is a prime factor of 500.
- 250 is the quotient.

2. Divide 250 by 2: 250 ÷ 2 = 125
- 2 is a prime factor of 500 (appears twice).
- 125 is the quotient.

3. Divide 125 by 5: 125 ÷ 5 = 25
- 5 is a prime factor of 500.
- 25 is the quotient.

4. Divide 25 by 5: 25 ÷ 5 = 5
- 5 is a prime factor of 500 (appears twice).
- 5 is the quotient.

5. Divide 5 by 5: 5 ÷ 5 = 1
- 5 is a prime factor of 500 (appears three times).
- 1 is the quotient.

Prime Factors of 500:
The prime factors of 500 are: 2^2 × 5^3.

Simplifying the Cube Root:
Now, let's simplify the cube root of 500 using the prime factors we found.

∛500 = ∛(2^2 × 5^3)

To simplify the cube root, we can take out any factor that appears thrice from inside the cube root.

Applying the Rationalizing Factor:
Since the prime factor 5 appears three times, we can take it out of the cube root.

∛500 = 5 × ∛(2^2 × 5)

Now, we can simplify the cube root further.

∛500 = 5 × ∛(4 × 5)

∛500 = 5 × ∛20

Therefore, the simplest rationalizing factor of the cube root of 500 is 5∛20.