Let's work through the questions from the image.

a) Write 900 as a product of its prime factors in index form.

We begin by performing prime factorization on 900.

1. Start by dividing 900 by the smallest prime number, 2: $900 \div 2 = 450$ $450 \div 2 = 225$ Since 225 is odd, it's no longer divisible by 2. Next, try 3: $225 \div 3 = 75$ $75 \div 3 = 25$ Now, 25 is not divisible by 3, so try 5: $25 \div 5 = 5$ $5 \div 5 = 1$

The prime factorization of 900 is $2^2 \times 3^2 \times 5^2$.

b) Use your answer from part a) to explain how you know that 900 is a square number.

From the prime factorization $900 = 2^2 \times 3^2 \times 5^2$, we can see that all the exponents of the prime factors are even.

A number is a square number if all the exponents in its prime factorization are even, because this ensures that the number can be expressed as the square of another integer. In this case: $900 = (2 \times 3 \times 5)^2 = 30^2$ Thus, 900 is a square number.

Would you like further details or have any other questions?

Here are 5 related questions:

1. How do you find the prime factorization of a number?
2. Why do even exponents in prime factorization indicate a square number?
3. What are the properties of square numbers?
4. How can you find the square root of a large number using prime factorization?
5. Can you explain the difference between prime factorization and factorization in general?

Tip: Always check if a number is divisible by the smallest primes (2, 3, 5, etc.) when performing prime factorization.