Find cube root of 110592.

To find the cube root of 110592, we can approach this problem step by step. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. In mathematical terms, if \( x \) is the cube root of \( y \), then \( x^3 = y \). So, we need to find a number \( x \) such that \( x^3 = 110592 \).

Breaking Down the Problem

One effective way to find the cube root is to factor the number into its prime factors. This method allows us to simplify the calculation significantly. Let's start by finding the prime factorization of 110592.

Finding Prime Factors

We can divide 110592 by the smallest prime numbers until we reach 1. Here’s how it goes:

• 110592 ÷ 2 = 55296
• 55296 ÷ 2 = 27648
• 27648 ÷ 2 = 13824
• 13824 ÷ 2 = 6912
• 6912 ÷ 2 = 3456
• 3456 ÷ 2 = 1728
• 1728 ÷ 2 = 864
• 864 ÷ 2 = 432
• 432 ÷ 2 = 216
• 216 ÷ 2 = 108
• 108 ÷ 2 = 54
• 54 ÷ 2 = 27
• 27 ÷ 3 = 9
• 9 ÷ 3 = 3
• 3 ÷ 3 = 1

From this process, we can see that:

110592 = 2^{11} × 3^3

Calculating the Cube Root

Now that we have the prime factorization, we can find the cube root. The cube root of a product of prime factors can be calculated by taking the cube root of each factor separately.

Using the property of exponents:

• Cube root of \( 2^{11} \) is \( 2^{11/3} \) which simplifies to \( 2^{3} \times 2^{2/3} \) or \( 8 \times 2^{2/3} \).
• Cube root of \( 3^{3} \) is \( 3^{3/3} = 3 \).

Now, we can combine these results:

Cube root of 110592 = \( 8 \times 3 \times 2^{2/3} \)

Approximating the Result

To find the cube root more precisely, we can calculate \( 2^{2/3} \). The cube root of 4 (which is \( 2^2 \)) is approximately 1.5874. Therefore:

Cube root of 110592 ≈ \( 8 \times 3 \times 1.5874 \approx 38.0992 \)

Final Answer

However, for practical purposes, we can round this to the nearest whole number. Thus, the cube root of 110592 is approximately 38.

In summary, through prime factorization and some calculations, we determined that the cube root of 110592 is about 38. This method not only gives us the answer but also helps reinforce the understanding of cube roots and prime factorization.