What is the cube root of 1024?

Steps to Solve

1. Identifying the Cube Root

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

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

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

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

4. 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:

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

5. Estimating the Remaining Value

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

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

6. Final Cube Root Estimate

Therefore, the approximate cube root of 1024 is around 10.08. The exact expression is:

$$ x = 8 \cdot \sqrt[3]{2} $$