https://file.aiquickdraw.com/mathbot/file/1727256846941z0n4ev7a.jpg

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.

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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.

Write 900 as a product of its prime factors in index form. Use your answer to explain how you know that 900 is a square number.

Understand how to express 900 as a product of its prime factors in index form and learn why it is a square number. This problem involves prime factorization and the properties of square numbers, suitable for students in Grades 6-8.

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