Express in index notation, 8960 as a product of its prime numbers.
To express 8960 as a product of its prime numbers in index notation, we need to determine its prime factors.
First, we can check if 2 is a factor of 8960. Since 8960 is an even number (ends in 0), we know that it is divisible by 2.
Dividing 8960 by 2, we get 4480.
Next, we check if 2 is still a factor of 4480.
Dividing 4480 by 2, we get 2240.
Continuing this process, we find that 2 is a factor of 8960 four times (2 x 2 x 2 x 2 = 16).
Next, we need to check the next prime number, 3.
Dividing 2240 by 3 is not possible without any remainder. Thus, 3 is not a factor of 8960.
Then, we can check the next prime number, 5.
Dividing 2240 by 5 gives us 448.
Further, dividing 448 by 5 gives us 89.6. However, 89.6 is not divisible by 5 as it has a decimal portion.
Therefore, 5 is not a factor of 8960.
Next, we can check the prime number 7.
Dividing 2240 by 7, we get 320.
Then, dividing 320 by 7 gives us 45.7142. However, 45.7142 is not divisible by 7 as it has a decimal portion.
Therefore, 7 is not a factor of 8960.
Lastly, we check the prime number 11.
Dividing 2240 by 11, we get 203.6363. However, 203.6363 is not divisible by 11 as it has a decimal portion.
Therefore, 11 is not a factor of 8960.
To express 8960 as a product of its prime numbers in index notation, we have:
8960 = 2^4 x 5^0 x 7^0 x 11^0
So, in index notation, 8960 is written as:
8960 = 2^4
To express 8960 as a product of its prime numbers using index notation, we need to find the prime factorization of 8960.
First, we divide 8960 by the smallest prime number, 2:
8960 ÷ 2 = 4480
We can divide 4480 by 2 again:
4480 ÷ 2 = 2240
Next, we continue dividing by 2:
2240 ÷ 2 = 1120
We can still divide by 2:
1120 ÷ 2 = 560
Now, let's divide by the next prime number, 5:
560 ÷ 5 = 112
Lastly, we divide by another prime, 2:
112 ÷ 2 = 56
We can still divide by 2:
56 ÷ 2 = 28
And again:
28 ÷ 2 = 14
Finally, we divide 14 by 2 one more time:
14 ÷ 2 = 7
Therefore, the prime factorization of 8960 is:
2^7 × 5 × 7
In index notation, this can be written as:
2^7 × 5^1 × 7^1
To express 8960 as a product of its prime numbers using index notation, we need to find the prime factorization of 8960. Here's how you can do it:
1. Divide 8960 by the smallest prime number, which is 2. Since 8960 is even, it is divisible by 2.
8960 ÷ 2 = 4480
2. Repeat step 1 until you can no longer divide evenly by 2.
4480 ÷ 2 = 2240
2240 ÷ 2 = 1120
1120 ÷ 2 = 560
560 ÷ 2 = 280
280 ÷ 2 = 140
140 ÷ 2 = 70
3. The result is no longer divisible by 2 evenly. Now, you move on to the next prime number, which is 3. Check if 8960 is divisible by 3.
70 ÷ 3 = 23.333...
4. Continue to the next prime number, which is 5.
70 ÷ 5 = 14
5. Divide by the next prime number, which is 7.
14 ÷ 7 = 2
Now, we have reached 2, which is a prime number and cannot be divided further. Therefore, the prime factorization of 8960 is:
2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 5 × 7 = 2^8 × 3 × 5 × 7
In index notation, we can write it as:
2^8 × 3^1 × 5^1 × 7^1
So, 8960 can be expressed in index notation as a product of its prime numbers as 2^8 × 3 × 5 × 7.