Find the cube root of -17576 using factorization:

To find the cube root of -17576 using factorization, we first need to factor the number into its prime components.

Step 1: Factor -17576

Since we are dealing with a negative number, we can express it as:

-17576 = -1 × 17576

Now, let's focus on factoring 17576.

Step 2: Prime Factorization of 17576

We can start dividing by small prime numbers:

• 17576 is even, so we divide by 2:
• 17576 ÷ 2 = 8788
• 8788 ÷ 2 = 4394
• 4394 ÷ 2 = 2197

Now, we need to factor 2197. We can check for cube roots:

Since 13 × 13 × 13 = 2197, we have:

2197 = 13³

Step 3: Combine the Factors

Putting it all together, we have:

-17576 = -1 × 2³ × 13³

Step 4: Calculate the Cube Root

The cube root of a product is the product of the cube roots:

∛(-17576) = ∛(-1) × ∛(2³) × ∛(13³)

This simplifies to:

∛(-1) = -1, ∛(2³) = 2, and ∛(13³) = 13.

Thus, we find:

∛(-17576) = -1 × 2 × 13 = -26.

Final Answer

The cube root of -17576 is -26.