A perfect square is a number that can be written as the product of two identical integers.
Yes, see solution.
Practice makes perfect
Recall that a perfect square is a number that can be written as the product of two identical integers or as a square power. Let's see a few examples!
Example
Rewrite as a Product
Rewrite as a Power
25
5 * 5
5^2
64
8 * 8
8^2
36
6 * 6
6^2
Consider the given prime factorization of a number.
2^4 * 3^4 * 5^4 * 7 ^2We want to know if the number is a perfect square. To do, let's rewrite each power as a product of repeated factors. Recall that the base of the power is the repeated factor and the exponent indicates the number of times the base is used as a factor.
Power
Repeated Factor
2^4
2 * 2 * 2 * 2
3^4
3 * 3 * 3 * 3
5^4
5 * 5 * 5 * 5
7^2
7 * 7
Note that we can rewrite the repeated factor as the product of two powers.
Power
Repeated Factor
Product of Powers
2^4
2 * 2 * 2 * 2
2^2 * 2^2
3^4
3 * 3 * 3 * 3
3^2 * 3^2
5^4
5 * 5 * 5 * 5
5^2 * 5^2
7^2
7 * 7
7^1 * 7^1
With this information, we can rewrite the given prime factorization as the product of powers.
2^2 * 2^2 * 3^2 * 3^2 * 5^2 * 5^2 * 7^1 * 7^1
Remember that we can change the order of the factors and the number will remain the same. This means that we can rewrite the product to get the product of two factors. The factor will be the product inside the parentheses.
(2^2 * 3^2 * 5^2 * 7^1) * (2^2 * 3^2 * 5^2 * 7^1)
Note that the product inside each parentheses is equal. This means that we have a product of two identical numbers. Therefore, the given number is a perfect square.
