Question
Express 120 as a product of prime numbers with exponents
2^3 x 3^2 x 5
2^3 x 3 x 5
2^2 x 3 x 5^2
2^2 x 3^2 x 5
2^3 x 3^2 x 5
2^3 x 3 x 5
2^2 x 3 x 5^2
2^2 x 3^2 x 5
Answers
GPT-4o mini
To express 120 as a product of prime numbers with exponents, we start by finding its prime factorization.
1. We can divide 120 by 2 (the smallest prime number):
- \( 120 ÷ 2 = 60 \)
- \( 60 ÷ 2 = 30 \)
- \( 30 ÷ 2 = 15 \)
(Now 15 is not divisible by 2 anymore.)
2. Next, we use the next smallest prime, which is 3:
- \( 15 ÷ 3 = 5 \)
3. Finally, 5 is already a prime number.
So, we can summarize our factorization:
- We divided by 2 three times (which gives \(2^3\)).
- We divided by 3 one time (which gives \(3^1\)).
- We also have 5, which is \(5^1\).
Putting it all together, we have:
\[
120 = 2^3 \times 3^1 \times 5^1
\]
This should match one of the formats listed in your question. However, it seems none of the options exactly match \(2^3 \times 3^1 \times 5^1\).
If we consider writing \(3^1\) and \(5^1\) with exponents (even though those exponents are 1), it is still valid to denote it simply as:
\[
120 = 2^3 \times 3^1 \times 5^1 = 2^3 \times 3 \times 5
\]
Among your options, the closest representation without needing to convert the 3 and 5 to exponents would be:
- \(2^3 \times 3 \times 5\)
So while it's not listed explicitly in the options you provided, you can reference \(2^3 \times 3^1 \times 5^1\) or \(2^3 \times 3 \times 5\).
If you are looking for just one of the options you initially provided, \(2^3 \times 3 \times 5\) is indeed shown in your list as:
- \(2^3 \times 3 \times 5\)
So the answer would be:
\[
2^3 x 3 x 5
\]
(Which technically corresponds to your option \(2^3 x 3^1 x 5^1\) where the 1s are not explicitly shown).
1. We can divide 120 by 2 (the smallest prime number):
- \( 120 ÷ 2 = 60 \)
- \( 60 ÷ 2 = 30 \)
- \( 30 ÷ 2 = 15 \)
(Now 15 is not divisible by 2 anymore.)
2. Next, we use the next smallest prime, which is 3:
- \( 15 ÷ 3 = 5 \)
3. Finally, 5 is already a prime number.
So, we can summarize our factorization:
- We divided by 2 three times (which gives \(2^3\)).
- We divided by 3 one time (which gives \(3^1\)).
- We also have 5, which is \(5^1\).
Putting it all together, we have:
\[
120 = 2^3 \times 3^1 \times 5^1
\]
This should match one of the formats listed in your question. However, it seems none of the options exactly match \(2^3 \times 3^1 \times 5^1\).
If we consider writing \(3^1\) and \(5^1\) with exponents (even though those exponents are 1), it is still valid to denote it simply as:
\[
120 = 2^3 \times 3^1 \times 5^1 = 2^3 \times 3 \times 5
\]
Among your options, the closest representation without needing to convert the 3 and 5 to exponents would be:
- \(2^3 \times 3 \times 5\)
So while it's not listed explicitly in the options you provided, you can reference \(2^3 \times 3^1 \times 5^1\) or \(2^3 \times 3 \times 5\).
If you are looking for just one of the options you initially provided, \(2^3 \times 3 \times 5\) is indeed shown in your list as:
- \(2^3 \times 3 \times 5\)
So the answer would be:
\[
2^3 x 3 x 5
\]
(Which technically corresponds to your option \(2^3 x 3^1 x 5^1\) where the 1s are not explicitly shown).