Check whether 90 is a perfect square or not by using prime factorisation.

Answer

To determine whether 90 is a perfect square using prime factorization, we can follow these steps:

### Step 1: Prime Factorization of 90
We start by finding the prime factors of 90.

1. The smallest prime number is 2. Since 90 is even, we divide by 2:
\[ 90 \div 2 = 45 \]
So, we have one factor of 2.

2. Next, we factor 45. The smallest prime number that divides 45 is 3:
\[ 45 \div 3 = 15 \]
So, we have one factor of 3.

3. We continue factoring 15. Again, the smallest prime number is 3:
\[ 15 \div 3 = 5 \]
So, we have another factor of 3.

4. Finally, we factor 5, which is a prime number itself:
\[ 5 \div 5 = 1 \]
So, we have one factor of 5.

Thus, the prime factorization of 90 is:
\[ 90 = 2^1 \times 3^2 \times 5^1 \]

### Step 2: Analyzing the Prime Factorization
To check if 90 is a perfect square, we need to look at the exponents of the prime factors in its prime factorization.

- The exponent of 2 is 1.
- The exponent of 3 is 2.
- The exponent of 5 is 1.

### Step 3: Checking for Perfect Square Conditions
A number is a perfect square if all the exponents in its prime factorization are even.

- The exponent of 2 (1) is odd.
- The exponent of 3 (2) is even.
- The exponent of 5 (1) is odd.

Since not all the exponents are even, we conclude that:

### Conclusion
90 is **not** a perfect square.

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