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25 votes
25 votes
1458 smallest whole number by which it should be multiplied so as to get a perfect square number.also find the square root of the square number obtained

User Pragati Sureka
by
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2 Answers

11 votes
11 votes

Final answer:

The smallest whole number to multiply 1458 by to get a perfect square is 6, and the square root of the resulting square number (8748) is 54.

Step-by-step explanation:

The question seeks to find the smallest whole number by which 1458 should be multiplied to obtain a perfect square. To approach this, we need to factorize 1458 and determine its prime factors. We'll then identify which factors need to be squared to create a perfect square and thus find the smallest multiplier.

First, let's factor 1458:

1458 = 2 × 3 × 3 × 3 × 3 × 3 × 3.

We can see that 1458 has a prime factorization with one factor of 2 and six factors of 3. To get a perfect square, each prime factor must appear an even number of times, so we need to multiply 1458 by another 2 to even out the number of 2's and by one more 3 to make the number of 3's even. Therefore, we multiply by 2 × 3 = 6.

The resulting perfect square is 1458 × 6 = 8748.

To find the square root of the new perfect square (8748), we pair the prime factors:

8748 = 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3.

Taking the square root gives us 2 × 3 × 3 × 3 = 54.

Hence, the square root of the square number obtained is 54.

User HHeckner
by
2.9k points
17 votes
17 votes

9514 1404 393

Answer:

  • 2
  • 54

Step-by-step explanation:

The prime factorization of 1458 is ...

1458 = 2·3^6

The exponent of 3 is even, but we need another factor of 2 to make the exponent of that factor even. You will get a perfect square by multiplying by 2.

√(2·1458) = √2916 = 54

Multiplying by 2 will give the square of 54.

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