Question

the number $345{,}600$ can be expressed as $6^a5^b4^c$ for integers $a$, $b$ and $c.$ what is the value of the product $abc?$

Answer 1

The product $abc$ for the expression $345{,}600 = 6^a5^b4^c$ is $abc = \underline{4} \times \underline{4} \times \underline{3} = \underline{48}.$

Step-by-step explanation:

The given number is $345{,}600,$ and we need to express it in the form $6^a5^b4^c.$ Let's break down the given number into its prime factors:

`\[345{,}600 = 2^6 * 3^4 * 5^2 * 7^2.\]`

Now, we want to express this in terms of $6^a5^b4^c.$ Notice that $6 =
`2 * 3$ and $4 = 2^2.$` So, we can rewrite the prime factorization using these bases:

`\[345{,}600 = (2^2)^3 * 3^4 * 5^2 * 7^2 = 4^3 * 3^4 * 5^2 * 7^2.\]`

Comparing this with $6^a5^b4^c,$ we find that $a = 3,$ $b = 2,$ and $c = 4.$ Now, the product $abc$ is simply $3 \times 2 \times 4 = 48,$ which is our final answer.

In summary, the prime factorization of $345{,}600$ expressed in terms of $6^a5^b4^c$ is found by recognizing the relationship between $6,$ $5,$ and $4$ with their prime factors. Once we have the values of $a,$ $b,$ and $c,$ the product $abc$ gives us the desired result, which is $48.$

Answer 2

To express the number 345,600 as a product of powers of 6, 5, and 4, divide 345,600 by each prime factor until you can no longer divide evenly. The product of the exponents will give the value of the product abc.

Step-by-step explanation:

To express the number 345,600 as a product of powers of 6, 5, and 4, we need to determine the exponents a, b, and c for each prime factor. We can divide 345,600 by 6 repeatedly until we can no longer divide evenly. We can divide the result by 5 next, and then by 4. The exponents will be the number of times we successfully divided by each prime factor.

345,600 ÷ 6 = 57,600 ÷ 6 = 9,600 ÷ 6 = 1,600 ÷ 6 = 266.67 ≈ 266

266 ÷ 5 = 53.2 ≈ 53

53 ÷ 4 = 13.25 ≈ 13

Therefore, the exponents a, b, and c are 4, 2, and 1, respectively. So, the product abc is 4 × 2 × 1 = 8.