³√128:³√2000

To solve this, you'll need to calculate the cube root of each number and then divide them.

The cube root of 128 (∛128) can be estimated by recognizing that 4³ = 64 and 5³ = 125. Since 128 is closer to 125, ∛128 is a bit more than 5, but for the exact value, you can either use a calculator or simplify it to 2∛2, as follows:

128 = 2^7 = 2^6 × 2 = (2^3)² × 2 = 8² × 2 = 64 × 2 Thus, ∛128 = ∛(64×2) = ∛(8²×2) = ∛(4³×2) = ∛(4×4×4×2) = 4∛2

For the cube root of 2000 (∛2000), notice that 12³ = 1728 and 13³ = 2197, so the cube root of 2000 must lie between 12 and 13. For the exact value, you can use a calculator or approximate it as follows:

2000 = 2^4 × 5^3 = (2×2×2×2) × (5×5×5) Recognizing that 8³ = 512 and 10³ = 1000 are not very close, it is more useful to use a calculator for the precise number, but we can still say that ∛2000 is somewhat closer to 12 than it is to 13.

So, dividing ∛128 by ∛2000, if we were to use the approximate simplifications, would involve dividing 4∛2 by a number somewhat above 12. However, without precise numbers, this would be an estimate and not very accurate. For exact values, it's best to use a calculator. If we do use one, we get:

∛128 ≈ 5.0397 ∛2000 ≈ 12.5992

So the division of ∛128:∛2000 would be approximately:

5.0397 ÷ 12.5992 ≈ 0.4

This is only an estimate, and for precise calculations, you would want to use the actual values derived from a calculator.