Given that 240 equals two to the power of 4×3×5 And y Equals 240×252 right why as a product of its prime factors

To express \( y \) as a product of its prime factors, we first need to break down the numbers 240 and 252 into their prime factors.

### Step 1: Prime factorization of 240
240 is already given as \( 2^4 \times 3 \times 5 \). This means:
\[
240 = 2^4 \times 3 \times 5
\]

### Step 2: Prime factorization of 252
Now, let's find the prime factorization of 252.

- Start by dividing 252 by 2 (the smallest prime number):
\[
252 \div 2 = 126
\]
- Divide 126 by 2:
\[
126 \div 2 = 63
\]
- 63 is divisible by 3:
\[
63 \div 3 = 21
\]
- Divide 21 by 3:
\[
21 \div 3 = 7
\]

So, the prime factorization of 252 is:
\[
252 = 2^2 \times 3^2 \times 7
\]

### Step 3: Multiply 240 and 252 and express as product of prime factors
Now, multiply the factorizations of 240 and 252:
\[
y = 240 \times 252 = (2^4 \times 3 \times 5) \times (2^2 \times 3^2 \times 7)
\]

- Combine the powers of 2:
\[
2^4 \times 2^2 = 2^6
\]
- Combine the powers of 3:
\[
3 \times 3^2 = 3^3
\]
- Then, include the other primes (5 and 7).

So, \( y \) as a product of its prime factors is:
\[
y = 2^6 \times 3^3 \times 5 \times 7
\]

### Final Answer:
\[
y = 2^6 \times 3^3 \times 5 \times 7
\]

Would you like more details on prime factorization or any part of the process?

Here are five related questions to further expand your understanding:
1. How do you find the prime factorization of a large number efficiently?
2. Can you explain why the fundamental theorem of arithmetic guarantees unique prime factorizations?
3. How does the LCM (least common multiple) relate to prime factorizations?
4. How would you calculate the greatest common divisor (GCD) using prime factorization?
5. What is the significance of powers in prime factorizations when solving algebraic equations?

**Tip:** Always break down numbers step-by-step when doing prime factorization. This ensures accuracy and helps in understanding the structure of the number!

Learn how to break down 240 and 252 into their prime factors and multiply them step by step. This problem illustrates key mathematical concepts such as prime factorization and the fundamental theorem of arithmetic. Suitable for students in Grades 8-10.

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