Express in index notation, 8960 as a product of its prime factors
The prime factorization of 8960 can be written as:
8960 = 2^7 * 5 * 7
To express 8960 as a product of its prime factors, we need to find the prime numbers that divide evenly into 8960.
Let's start by dividing 8960 by the smallest prime number, 2.
8960 ÷ 2 = 4480
Now we continue dividing by 2 until we can no longer divide evenly:
4480 ÷ 2 = 2240
2240 ÷ 2 = 1120
1120 ÷ 2 = 560
560 ÷ 2 = 280
280 ÷ 2 = 140
140 ÷ 2 = 70
70 ÷ 2 = 35
Now, let's move on to the next prime number, 3, and continue dividing until we can no longer divide evenly:
35 ÷ 3 = 11.67 (not divisible by 3)
The largest prime factor of 8960 is 2, and we have already expressed it as a product of 2s:
8960 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 5 × 7
So, in index notation, 8960 can be expressed as:
8960 = 2^11 × 3 × 5 × 7
To express a number as a product of its prime factors, you need to find the prime numbers that divide evenly into the given number.
To find the prime factors of 8960, you start by dividing it by the smallest prime number, which is 2. If 8960 is divisible by 2, then 2 is a prime factor. If it is not divisible by 2, you move on to the next prime number, which is 3.
Let's check if 8960 is divisible evenly by 2:
8960 ÷ 2 = 4480
Since 4480 is divisible by 2, we can continue:
4480 ÷ 2 = 2240
Again, we divide by 2:
2240 ÷ 2 = 1120
Now, we repeat the process:
1120 ÷ 2 = 560
And once again:
560 ÷ 2 = 280
Next, we try dividing by 2 again:
280 ÷ 2 = 140
Now, we reach a point where we can no longer divide by 2. So, we move on to the next prime number, which is 3.
Let's check if 140 is divisible evenly by 3:
140 ÷ 3 = 46 remainder 2
Since 140 is not divisible evenly by 3, we move on to the next prime number, which is 5.
Let's check if 140 is divisible evenly by 5:
140 ÷ 5 = 28
Since 140 is divisible evenly by 5, we continue:
28 ÷ 5 = 5 remainder 3
Now, we check if 28 is divisible by 5:
28 ÷ 5 = 5 remainder 3
We can see that 28 is not divisible evenly by 5 either. So, we move on to the next prime number, which is 7.
Let's check if 28 is divisible evenly by 7:
28 ÷ 7 = 4
Since 28 is divisible evenly by 7, we continue:
4 ÷ 7 = 4
Now, we can see that we cannot divide by 7 evenly either. This means we have found all the prime factors of 8960.
Therefore, in index notation, 8960 can be expressed as a product of its prime factors as:
2^7 × 5 × 7
Here, the "^" symbol represents exponentiation, so 2^7 means 2 raised to the power of 7.