Answer :
To determine whether the number 1953125 is a square number, we need to understand the properties of square numbers and how they relate to their prime factorization.
1. Definition of a Square Number:
A square number, or perfect square, is an integer that can be expressed as the square of another integer. In mathematical terms, if [tex]\( n \)[/tex] is a square number, then there exists an integer [tex]\( k \)[/tex] such that [tex]\( n = k^2 \)[/tex].
2. Prime Factorization and Exponents:
When a number is expressed in terms of its prime factors, each exponent in the prime factorization must be even for the number to be a perfect square. This is because squaring an integer multiplies each of its prime exponent values by 2, resulting in all exponents being even.
3. Given Information:
We are given that the number 1953125 can be expressed as [tex]\( 5^9 \)[/tex]. This means that 1953125 is the result of raising the prime number 5 to the power of 9.
4. Analyzing the Exponent:
In the prime factorization [tex]\( 5^9 \)[/tex], the exponent is 9. For 1953125 to be a perfect square, all the exponents in its prime factorization must be even. However, 9 is an odd number. Since the exponent 9 is not even, this already tells us that 1953125 cannot be a perfect square.
5. Conclusion:
Because we have established that for a number to be a perfect square, the exponents in its prime factorization must all be even—and since 9 is an odd exponent—the number 1953125 cannot be a perfect square. Thus, we can confidently conclude that 1953125 is not a square number.
In summary, the number 1953125 is equal to [tex]\( 5^9 \)[/tex]. Since 9 is an odd number and not even, this indicates that 1953125 is not a square number.
1. Definition of a Square Number:
A square number, or perfect square, is an integer that can be expressed as the square of another integer. In mathematical terms, if [tex]\( n \)[/tex] is a square number, then there exists an integer [tex]\( k \)[/tex] such that [tex]\( n = k^2 \)[/tex].
2. Prime Factorization and Exponents:
When a number is expressed in terms of its prime factors, each exponent in the prime factorization must be even for the number to be a perfect square. This is because squaring an integer multiplies each of its prime exponent values by 2, resulting in all exponents being even.
3. Given Information:
We are given that the number 1953125 can be expressed as [tex]\( 5^9 \)[/tex]. This means that 1953125 is the result of raising the prime number 5 to the power of 9.
4. Analyzing the Exponent:
In the prime factorization [tex]\( 5^9 \)[/tex], the exponent is 9. For 1953125 to be a perfect square, all the exponents in its prime factorization must be even. However, 9 is an odd number. Since the exponent 9 is not even, this already tells us that 1953125 cannot be a perfect square.
5. Conclusion:
Because we have established that for a number to be a perfect square, the exponents in its prime factorization must all be even—and since 9 is an odd exponent—the number 1953125 cannot be a perfect square. Thus, we can confidently conclude that 1953125 is not a square number.
In summary, the number 1953125 is equal to [tex]\( 5^9 \)[/tex]. Since 9 is an odd number and not even, this indicates that 1953125 is not a square number.
