What is the cube root of 27000?
Understand the Problem
The question is asking for the cube root of the number 27000, which involves finding a number that, when multiplied by itself three times, equals 27000.
Answer
$30$
Answer for screen readers
The cube root of 27000 is $30$.
Steps to Solve
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Expression of the cube root
To find the cube root of a number, we can express it mathematically as:
$$ \sqrt[3]{27000} $$ -
Prime factorization of 27000
Next, we will perform the prime factorization of 27000 to find its prime factors.
27000 can be divided by 10 to simplify the process:
$$ 27000 = 27 \times 1000 = 27 \times 10^3 $$
Now, let's factor each component:
$$ 27 = 3^3 \quad \text{and} \quad 1000 = 10^3 = (2 \times 5)^3 = 2^3 \times 5^3 $$
Combining these, we find:
$$ 27000 = 3^3 \times 2^3 \times 5^3 $$ -
Finding cube root of the factors
Now, we can take the cube root of each component:
$$ \sqrt[3]{27000} = \sqrt[3]{3^3 \times 2^3 \times 5^3} = \sqrt[3]{3^3} \times \sqrt[3]{2^3} \times \sqrt[3]{5^3} $$
This simplifies to:
$$ 3 \times 2 \times 5 $$ -
Final multiplication
Finally, multiplying the results together gives:
$$ 3 \times 2 \times 5 = 30 $$
The cube root of 27000 is $30$.
More Information
The cube root of 27000 is significant as it's often used in geometry and volume calculations. Understanding cube roots helps in solving problems involving three-dimensional measurements.
Tips
- Forgetting to factor the number correctly, leading to an incorrect prime factorization.
- Not simplifying the cube root expression properly, which can result in errors in multiplication.
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