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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

  1. Identifying the Cube Root

To find the cube root of 1024, we need to express it as an equation:

$$ x = \sqrt[3]{1024} $$

  1. 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: $$ 1024 = 2^{10} $$

  1. 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: $$ x = 2^{10/3} $$

  1. 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:

$$ x = 8 \cdot 2^{1/3} $$

  1. Estimating the Remaining Value

The remaining value is $2^{1/3}$, which is the cube root of 2, approximately 1.26. Thus:

$$ x \approx 8 \cdot 1.26 \approx 10.08 $$

  1. 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.
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