What is the cube root of 64000?

Steps to Solve

1. Break down the number into prime factors

To find the cube root of 64000, we should first express 64000 as a product of its prime factors.

$$ 64000 = 64 \times 1000 $$

Next, we factor these numbers:

$$ 64 = 2^6 \quad \text{and} \quad 1000 = 10^3 = (2 \times 5)^3 = 2^3 \times 5^3 $$

So,

$$ 64000 = 2^6 \times (2^3 \times 5^3) = 2^{6+3} \times 5^3 = 2^9 \times 5^3 $$

2. Use the property of cube roots

We can now apply the property of cube roots to the prime factorization. The cube root of a product is the product of the cube roots:

$$ \sqrt[3]{64000} = \sqrt[3]{2^9 \times 5^3} $$

3. Simplify each factor

Now we can simplify the cube root:

$$ \sqrt[3]{2^9} \times \sqrt[3]{5^3} $$

Calculating each piece:

• For ( \sqrt[3]{2^9} ):

$$ \sqrt[3]{2^9} = 2^{9/3} = 2^3 = 8 $$

• For ( \sqrt[3]{5^3} ):

$$ \sqrt[3]{5^3} = 5^{3/3} = 5^1 = 5 $$

4. Combine the results

Now we combine the simplified results:

$$ \sqrt[3]{64000} = 8 \times 5 = 40 $$