Understanding Prime Factorization

To express a number like 486 as a product of powers of prime factors, we need to find the prime numbers that multiply together to give 486. This process is called prime factorization.

A prime number is a whole number greater than 1 that has only two factors: 1 and itself. Examples include 2, 3, 5, 7, 11, and so on.

Finding the Prime Factors of 486

We can find the prime factors of 486 by repeatedly dividing it by the smallest possible prime number until we reach 1. Let's start with the smallest prime number, 2.

• 486 is an even number, so it is divisible by 2.
• $486 \div 2 = 243$
• Now consider 243. It's not divisible by 2 (it's odd). Let's try the next prime number, 3.
• To check if 243 is divisible by 3, we can sum its digits: $2+4+3 = 9$. Since 9 is divisible by 3, 243 is also divisible by 3.
• $243 \div 3 = 81$
• Consider 81. $8+1 = 9$, which is divisible by 3.
• $81 \div 3 = 27$
• Consider 27. $2+7 = 9$, which is divisible by 3.
• $27 \div 3 = 9$
• Consider 9. 9 is divisible by 3.
• $9 \div 3 = 3$
• Consider 3. 3 is a prime number, so it is divisible by 3.
• $3 \div 3 = 1$

We have reached 1, so the prime factorization is complete. The prime factors we used are 2 and 3, with 3 appearing multiple times.

Expressing 486 as a Product of Prime Factors

From the division steps, we can write 486 as the product of its prime factors:

$486 = 2 \times 3 \times 3 \times 3 \times 3 \times 3$

Expressing the Product Using Powers

To express this product using powers (exponents), we count how many times each prime factor appears:

• The factor 2 appears once. So, we write it as $2^1$.
• The factor 3 appears five times. So, we write it as $3^5$.

Therefore, the prime factorization of 486 expressed as a product of powers of prime factors is:

$\qquad 486 = 2^1 \times 3^5$

Comparing with Options

Let's compare our result $2^1 \times 3^5$ with the given options:

• Option 1: $5^3 \times 2 = 125 \times 2 = 250$
• Option 2: $2^1 \times 3^5 = 2 \times (3 \times 3 \times 3 \times 3 \times 3) = 2 \times 243 = 486$
• Option 3: $1^2 \times 5^3 = 1 \times 125 = 125$ (Note: 1 is not a prime factor)
• Option 4: $3^5 = 243$

Our calculated result, $2^1 \times 3^5$, matches Option 2 and is equal to 486.

Thus, 486 expressed as a product of powers of prime factors is $2^1 \times 3^5$.