Find the smallest number that should be multiplied with 54000 to make it a perfect cube?3524

1. Break Down the Problem

To find the smallest number to multiply with 54000 so it becomes a perfect cube, we first need to perform the prime factorization of 54000.

2. Relevant Concepts

A perfect cube is a number that can be expressed as $n^3$, where $n$ is an integer. For any prime factorization, each prime factor's exponent must be a multiple of 3 to form a perfect cube.

3. Analysis and Detail

1. Prime Factorization of 54000:

   • Break down 54000:

   $54000 = 54 \times 1000$ $54 = 2 \times 27 = 2 \times 3^3$ $1000 = 10^3 = (2 \times 5)^3 = 2^3 \times 5^3$ Thus, combining all together: $54000 = 2^1 \times 3^3 \times 2^3 \times 5^3 = 2^{1+3} \times 3^3 \times 5^3 = 2^4 \times 3^3 \times 5^3$

2. Identify the exponents: $54000 = 2^4 \times 3^3 \times 5^3$

   • The exponents are $4$ (for $2$), $3$ (for $3$), and $3$ (for $5$).

3. Make each exponent a multiple of 3:

   • For $2^4$, we need $2^{3-4} = 2^2$ (which is $4$) to make it $2^6$.
   • For $3^3$, it's already $3$.
   • For $5^3$, it's also already $3$.

   Hence, we need to multiply by $2^2$ to make the exponent of $2$ a multiple of 3.

4. Verify and Summarize

Multiplying $54000$ by $4$ (which is $2^2$) will adjust the exponent of $2$ to $6$:

• The new prime factorization will be: $54000 \times 4 = 2^{4+2} \times 3^3 \times 5^3 = 2^6 \times 3^3 \times 5^3$

This makes all factors' exponents multiples of $3$, confirming that our multiplication yields a perfect cube.

Final Answer

The smallest number that should be multiplied with 54000 to make it a perfect cube is $4$.