Find the smallest number with which is of the following numbers must be divided to get a perfect cube find the cube root of the number does obtained 145800 short solution 

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.

Find the smallest number with which the following numbers must be divided to get a perfect cube. Find the cube root of the number obtained: 145800.

Learn how to find the smallest number by which 145800 must be divided to get a perfect cube and how to calculate its cube root. This solution covers prime factorization and cube root calculation, suitable for Grades 9-12.

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