What is the cube root of 64000?
Understand the Problem
The question is asking to find the cube root of 64000, which is a mathematical operation that requires determining what number multiplied by itself three times equals 64000.
Answer
The cube root of 64000 is $40$.
Answer for screen readers
The cube root of 64000 is $40$.
Steps to Solve
- 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,
$$ 64000 = 2^6 \times (2^3 \times 5^3) = 2^{6+3} \times 5^3 = 2^9 \times 5^3 $$
- 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:
$$ \sqrt[3]{64000} = \sqrt[3]{2^9 \times 5^3} $$
- 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} ):
$$ \sqrt[3]{5^3} = 5^{3/3} = 5^1 = 5 $$
- Combine the results
Now we combine the simplified results:
$$ \sqrt[3]{64000} = 8 \times 5 = 40 $$
The cube root of 64000 is $40$.
More Information
Finding cube roots can be useful in various applications such as geometry, architecture, and even engineering when dealing with volumes of cubes or other cubic shapes.
Tips
- Not fully factoring the number into prime factors, which could lead to incorrect calculations.
- Forgetting to apply the cube root properties correctly when combining the factors.