We are given the number 145800, and the goal is to find the smallest number by which it should be divided to make it a perfect cube. After that, we need to find the cube root of the resulting number.

Step 1: Prime factorization of 145800

First, let's perform the prime factorization of 145800.

145800 is divisible by 2: $145800 \div 2 = 72900$ $72900 \div 2 = 36450$ $36450 \div 2 = 18225$ 18225 is not divisible by 2, but it is divisible by 3: $18225 \div 3 = 6075$ $6075 \div 3 = 2025$ $2025 \div 3 = 675$ $675 \div 3 = 225$ $225 \div 3 = 75$ 75 is divisible by 5: $75 \div 5 = 15$ $15 \div 5 = 3$ $3 \div 3 = 1$

Thus, the prime factorization of 145800 is: $145800 = 2^3 \times 3^6 \times 5^2$

Step 2: Adjusting to make a perfect cube

For a number to be a perfect cube, all the exponents in its prime factorization must be multiples of 3. Let's examine the exponents:

• The exponent of 2 is 3 (already a multiple of 3).
• The exponent of 3 is 6 (already a multiple of 3).
• The exponent of 5 is 2 (not a multiple of 3).

To make the exponent of 5 a multiple of 3, we need to divide the number by 5^2 = 25.

Step 3: Resulting perfect cube

Now, divide 145800 by 25: $145800 \div 25 = 5832$

Step 4: Cube root of 5832

Next, find the cube root of 5832: $\sqrt[3]{5832} = 18$

Final Answer:

• The smallest number by which 145800 must be divided to get a perfect cube is 25.
• The cube root of the resulting number is 18.

Would you like more details on any step? Here are some related questions to explore further:

1. What is the process for prime factorization of any number?
2. How do you find the cube root of a number manually?
3. Why must all exponents be multiples of 3 for a number to be a perfect cube?
4. What are the properties of cubes of numbers?
5. How do you determine the smallest number to divide to achieve a perfect square or cube?

Tip: When factorizing numbers, always start with the smallest primes like 2, 3, and 5 for efficiency.