To solve the problem of finding the least number that should be multiplied by [tex]\(2 \times 2 \times 7 \times 7 \times 5 \times 7 \times 5 \times 5\)[/tex] to make it a perfect cube, follow these steps:
1. Prime Factorization: List out the prime factors of the given number:
[tex]\[
2 \times 2 \times 7 \times 7 \times 5 \times 7 \times 5 \times 5 = 2^2 \times 7^3 \times 5^3
\][/tex]
2. Count the Exponents: Determine the exponents of each prime factor:
- For [tex]\(2\)[/tex]: The exponent is 2.
- For [tex]\(7\)[/tex]: The exponent is 3.
- For [tex]\(5\)[/tex]: The exponent is 3.
3. Perfect Cube Condition: For a number to be a perfect cube, each of the exponents in its prime factorization must be a multiple of 3.
4. Adjust the Exponents:
- The exponent of [tex]\(2\)[/tex] is 2, which is not a multiple of 3. To make it a multiple of 3, we need to multiply by one more [tex]\(2\)[/tex] (since [tex]\(2 + 1 = 3\)[/tex]).
- The exponents for both [tex]\(7\)[/tex] and [tex]\(5\)[/tex] are already multiples of 3, so no adjustments are needed for these primes.
5. Determine the Least Number: The least number we need to multiply by is the number that contains the missing prime factor raised to the appropriate power to make all the exponents multiples of 3. From step 4, this number is [tex]\(2\)[/tex].
Therefore, the least number that should be multiplied by [tex]\(2 \times 2 \times 7 \times 7 \times 5 \times 7 \times 5 \times 5\)[/tex] to make it a perfect cube is:
[tex]\[
\boxed{2}
\][/tex]
