How do you simplify ³√432?

To simplify the cube root of 432, we first need to factor the number into its prime components.

Step 1: Prime Factorization

Let's break down 432:

• 432 is even, so we can divide by 2: 432 ÷ 2 = 216
• 216 is also even: 216 ÷ 2 = 108
• 108 is even again: 108 ÷ 2 = 54
• 54 is even: 54 ÷ 2 = 27
• 27 is divisible by 3: 27 ÷ 3 = 9
• 9 is also divisible by 3: 9 ÷ 3 = 3
• Finally, 3 ÷ 3 = 1

So, the prime factorization of 432 is:

432 = 2^4 × 3^3

Step 2: Applying the Cube Root

Now, we can simplify the cube root:

³√432 = ³√(2^4 × 3^3)

Using the property of cube roots, we can separate the factors:

³√432 = ³√(2^4) × ³√(3^3)

Step 3: Simplifying Each Part

Next, we simplify each cube root:

• ³√(2^4) = 2^(4/3) = 2^(1 + 1/3) = 2 × ³√2
• ³√(3^3) = 3

Final Result

Combining these results gives us:

³√432 = 3 × 2 × ³√2 = 6 × ³√2

Thus, the simplified form of ³√432 is 6 × ³√2.