What is the cube root of 1024?
Understand the Problem
The question is asking for the cube root of the number 1024. This involves determining which number, when multiplied by itself three times, equals 1024.
Answer
The cube root of 1024 is approximately $10.08$.
Answer for screen readers
The cube root of 1024 is approximately $10.08$.
Steps to Solve
- Identifying the Cube Root
To find the cube root of 1024, we need to express it as an equation:
$$ x = \sqrt[3]{1024} $$
- Finding Prime Factorization
Next, we can factor 1024 into its prime factors.
1024 can be divided by 2: $$ 1024 \div 2 = 512 $$ $$ 512 \div 2 = 256 $$ $$ 256 \div 2 = 128 $$ $$ 128 \div 2 = 64 $$ $$ 64 \div 2 = 32 $$ $$ 32 \div 2 = 16 $$ $$ 16 \div 2 = 8 $$ $$ 8 \div 2 = 4 $$ $$ 4 \div 2 = 2 $$ $$ 2 \div 2 = 1 $$
Thus, the prime factorization of 1024 is: $$ 1024 = 2^{10} $$
- Applying the Cube Root to the Factorization
Now that we have the prime factorization, we apply the cube root:
$$ x = \sqrt[3]{2^{10}} $$ Using the property of exponents: $$ x = 2^{10/3} $$
- Simplifying the Exponent
Now, we simplify $2^{10/3}$ by breaking it down:
$$ 2^{10/3} = 2^{3 + 1/3} = 2^3 \cdot 2^{1/3} $$ Calculating $2^3$ gives us 8:
$$ x = 8 \cdot 2^{1/3} $$
- Estimating the Remaining Value
The remaining value is $2^{1/3}$, which is the cube root of 2, approximately 1.26. Thus:
$$ x \approx 8 \cdot 1.26 \approx 10.08 $$
- Final Cube Root Estimate
Therefore, the approximate cube root of 1024 is around 10.08. The exact expression is:
$$ x = 8 \cdot \sqrt[3]{2} $$
The cube root of 1024 is approximately $10.08$.
More Information
The cube root represents the number that, when multiplied by itself three times, equals the original number. In this case, $10.08$ is not a perfect integer, which indicates 1024 is not a perfect cube.
Tips
- Miscalculating the prime factorization of 1024 could lead to incorrect cube root values. Always double-check division steps.
- Forgetting to simplify the expression $2^{10/3}$ can lead to incorrect final answers. Always break down the exponent fully.