What is the cube root of 64000?

Understand the Problem

The question is asking for the cube root of 64000. This involves finding a number that, when multiplied by itself three times (cubed), equals 64000. To solve it, we'll calculate the cube root of the number.

Answer

$40$

Steps to Solve

Express the number in prime factorization

To find the cube root, we first find the prime factorization of 64000.

Starting with 64000: $$ 64000 = 64 \times 1000 $$

Breaking it down further: $$ 64 = 2^6 \quad \text{and} \quad 1000 = 10^3 = (2 \times 5)^3 = 2^3 \times 5^3 $$

Now, combining these gives us: $$ 64000 = 2^6 \times 2^3 \times 5^3 = 2^{6+3} \times 5^3 = 2^9 \times 5^3 $$

Finding the cube root using the prime factors

Now that we have the prime factorization: $$ 64000 = 2^9 \times 5^3 $$

We take the cube root of each factor: $$ \sqrt[3]{64000} = \sqrt[3]{2^9} \times \sqrt[3]{5^3} $$

Simplifying the cube roots

Using the properties of exponents: $$ \sqrt[3]{2^9} = 2^{9/3} = 2^3 = 8 $$ $$ \sqrt[3]{5^3} = 5^{3/3} = 5^1 = 5 $$

Combining the results

Now we multiply the results of the cube roots: $$ \sqrt[3]{64000} = 2^3 \times 5 = 8 \times 5 = 40 $$

The cube root of 64000 is $40$.

More Information

The cube root of a number is a value that, when multiplied by itself three times, results in that number. The prime factorization method is a powerful way to simplify calculations involving roots.

Tips

Forgetting to correctly break down the number into its prime factors.
Not applying properties of exponents correctly when simplifying the roots. To avoid this, always double-check your factorization and calculations.

AI-generated content may contain errors. Please verify critical information

Thank you for voting!