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:

  1. Perfect Square Condition:

    • A number is a perfect square if its exponent in the factorization is even.

  2. Perfect Cube Condition:

    • A number is a perfect cube if its exponent in the factorization is a multiple of three.

  3. 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.

  4. Factorization of 5:

    • The prime factorization of 5 is simply [tex]5^1[/tex].

  5. 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.

  6. Find 5 to the Power 5:

    • [tex]5^5 = 3125[/tex].

  7. 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.

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