Answer :
To determine the number by which 5 should be multiplied to make it both a perfect square and a perfect cube, we need to find a number that is a perfect sixth power. This is because a number that is both a perfect square and a perfect cube is a perfect sixth power.
Step-by-step explanation:
Perfect Square Condition:
- A number is a perfect square if its exponent in the factorization is even.
Perfect Cube Condition:
- A number is a perfect cube if its exponent in the factorization is a multiple of three.
Perfect Sixth Power Condition:
- A number is a perfect sixth power if it satisfies both the conditions above, meaning its exponent is a multiple of six.
Factorization of 5:
- The prime factorization of 5 is simply [tex]5^1[/tex].
Make it a Perfect Sixth Power:
- To make [tex]5^1[/tex] a perfect sixth power, we need to multiply it by [tex]5^5[/tex] so that the exponent becomes 6, since [tex]1 + 5 = 6[/tex].
- Therefore, [tex]5^1 \times 5^5 = 5^6[/tex] is a perfect sixth power.
Find 5 to the Power 5:
- [tex]5^5 = 3125[/tex].
Conclusion:
- Therefore, the number by which 5 should be multiplied to make it a perfect square and a perfect cube is 3125.
Thus, the multiple-choice option that makes 5 a perfect square and a perfect cube is 3125.
