Question
Express the the prime factorization 2^3 • 5^2 •11 as a whole number
Answers
GPT-4o mini
To express the prime factorization \(2^3 \cdot 5^2 \cdot 11\) as a whole number, we need to calculate the value of each part:
1. Calculate \(2^3\):
\[
2^3 = 2 \times 2 \times 2 = 8
\]
2. Calculate \(5^2\):
\[
5^2 = 5 \times 5 = 25
\]
3. Now multiply these results together with \(11\):
\[
8 \cdot 25 \cdot 11
\]
First, multiply \(8\) and \(25\):
\[
8 \cdot 25 = 200
\]
Next, multiply this result by \(11\):
\[
200 \cdot 11 = 2200
\]
Thus, the whole number represented by the prime factorization \(2^3 \cdot 5^2 \cdot 11\) is \(2200\).
1. Calculate \(2^3\):
\[
2^3 = 2 \times 2 \times 2 = 8
\]
2. Calculate \(5^2\):
\[
5^2 = 5 \times 5 = 25
\]
3. Now multiply these results together with \(11\):
\[
8 \cdot 25 \cdot 11
\]
First, multiply \(8\) and \(25\):
\[
8 \cdot 25 = 200
\]
Next, multiply this result by \(11\):
\[
200 \cdot 11 = 2200
\]
Thus, the whole number represented by the prime factorization \(2^3 \cdot 5^2 \cdot 11\) is \(2200\).
Express the prime factorization of 128 as a product of a prime numbers with exponents
GPT-4o mini
To express the prime factorization of \(128\), we start by determining its prime factorization.
We can divide \(128\) by \(2\) (the smallest prime number) repeatedly until we reach \(1\):
1. \(128 \div 2 = 64\)
2. \(64 \div 2 = 32\)
3. \(32 \div 2 = 16\)
4. \(16 \div 2 = 8\)
5. \(8 \div 2 = 4\)
6. \(4 \div 2 = 2\)
7. \(2 \div 2 = 1\)
Since we divided by \(2\) a total of \(7\) times before reaching \(1\), we can express \(128\) as:
\[
128 = 2^7
\]
Thus, the prime factorization of \(128\) is \(2^7\).
We can divide \(128\) by \(2\) (the smallest prime number) repeatedly until we reach \(1\):
1. \(128 \div 2 = 64\)
2. \(64 \div 2 = 32\)
3. \(32 \div 2 = 16\)
4. \(16 \div 2 = 8\)
5. \(8 \div 2 = 4\)
6. \(4 \div 2 = 2\)
7. \(2 \div 2 = 1\)
Since we divided by \(2\) a total of \(7\) times before reaching \(1\), we can express \(128\) as:
\[
128 = 2^7
\]
Thus, the prime factorization of \(128\) is \(2^7\).
Expressed the prime factorization of 54 as a product of a prime numbers with exponents
GPT-4o mini
To find the prime factorization of \(54\), we can start by dividing it by the smallest prime numbers:
1. Divide \(54\) by \(2\) (the smallest prime):
\[
54 \div 2 = 27
\]
This gives us \(2^1\) and \(27\) remaining.
2. Now, factor \(27\). It is divisible by \(3\) (the next smallest prime):
\[
27 \div 3 = 9
\]
This gives us \(3^1\) and \(9\) remaining.
3. Continue factoring \(9\):
\[
9 \div 3 = 3
\]
This gives us another \(3^1\) and \(3\) remaining.
4. Finally, factor the last \(3\):
\[
3 \div 3 = 1
\]
Now we can summarize the prime factorization:
- \(2^1\) for the factor of \(2\)
- \(3^3\) for the three factors of \(3\)
Putting it all together, we have:
\[
54 = 2^1 \cdot 3^3
\]
Thus, the prime factorization of \(54\) is \(2^1 \cdot 3^3\).
1. Divide \(54\) by \(2\) (the smallest prime):
\[
54 \div 2 = 27
\]
This gives us \(2^1\) and \(27\) remaining.
2. Now, factor \(27\). It is divisible by \(3\) (the next smallest prime):
\[
27 \div 3 = 9
\]
This gives us \(3^1\) and \(9\) remaining.
3. Continue factoring \(9\):
\[
9 \div 3 = 3
\]
This gives us another \(3^1\) and \(3\) remaining.
4. Finally, factor the last \(3\):
\[
3 \div 3 = 1
\]
Now we can summarize the prime factorization:
- \(2^1\) for the factor of \(2\)
- \(3^3\) for the three factors of \(3\)
Putting it all together, we have:
\[
54 = 2^1 \cdot 3^3
\]
Thus, the prime factorization of \(54\) is \(2^1 \cdot 3^3\).
Express the prime factorization of 144 as a product of prime factors by filling in the missing exponents after the caret
2^___ • 3^____=144
2^___ • 3^____=144
GPT-4o mini
To find the prime factorization of \(144\), we can start by dividing it by the smallest prime numbers:
1. First, divide \(144\) by \(2\) (the smallest prime):
\[
144 \div 2 = 72
\]
2. Divide \(72\) by \(2\):
\[
72 \div 2 = 36
\]
3. Divide \(36\) by \(2\):
\[
36 \div 2 = 18
\]
4. Divide \(18\) by \(2\):
\[
18 \div 2 = 9
\]
5. Now, divide \(9\) by \(3\):
\[
9 \div 3 = 3
\]
6. Finally, divide \(3\) by \(3\):
\[
3 \div 3 = 1
\]
Counting the factors, we have:
- \(2\) used \(4\) times.
- \(3\) used \(2\) times.
Thus, we can express the prime factorization of \(144\) as:
\[
144 = 2^4 \cdot 3^2
\]
Filling in the missing exponents, we have:
\[
2^4 \cdot 3^2 = 144
\]
So, the answer is:
- The exponent for \(2\) is \(4\).
- The exponent for \(3\) is \(2\).
1. First, divide \(144\) by \(2\) (the smallest prime):
\[
144 \div 2 = 72
\]
2. Divide \(72\) by \(2\):
\[
72 \div 2 = 36
\]
3. Divide \(36\) by \(2\):
\[
36 \div 2 = 18
\]
4. Divide \(18\) by \(2\):
\[
18 \div 2 = 9
\]
5. Now, divide \(9\) by \(3\):
\[
9 \div 3 = 3
\]
6. Finally, divide \(3\) by \(3\):
\[
3 \div 3 = 1
\]
Counting the factors, we have:
- \(2\) used \(4\) times.
- \(3\) used \(2\) times.
Thus, we can express the prime factorization of \(144\) as:
\[
144 = 2^4 \cdot 3^2
\]
Filling in the missing exponents, we have:
\[
2^4 \cdot 3^2 = 144
\]
So, the answer is:
- The exponent for \(2\) is \(4\).
- The exponent for \(3\) is \(2\).
Complete the factory tree with whole numbers to create the factor tree for 72
72
3•___
3•___
2•____
___•_____
72
3•___
3•___
2•____
___•_____
GPT-4o mini
To create a factor tree for \(72\), we can break it down into its prime factors.
1. Start with \(72\).
2. We can first factor \(72\) by dividing it by \(3\):
\[
72 = 3 \times 24
\]
3. Then, we take \(24\) and factor it further. \(24\) can be divided by \(2\):
\[
24 = 2 \times 12
\]
4. Next, we can factor \(12\) as:
\[
12 = 2 \times 6
\]
5. Finally, we have \(6\), which can be factored as:
\[
6 = 2 \times 3
\]
Now, we can fill in the factor tree:
```
72
/ \
3 24
/ \
2 12
/ \
2 6
/ \
2 3
```
So, the completed factor tree for \(72\) is as follows:
- **First level**: \(72 = 3 \times 24\)
- **Second level**: \(24 = 2 \times 12\)
- **Third level**: \(12 = 2 \times 6\)
- **Fourth level**: \(6 = 2 \times 3\)
Now, filling in the blanks in your tree with the whole numbers:
```
72
3•24
3•2•12
2•6
2•3
```
Thus, for your format:
- The first blank is \(24\).
- The second blank is \(2\).
- The third blank is \(12\).
- The fourth blank is \(6\).
- The fifth blank is \(3\).
1. Start with \(72\).
2. We can first factor \(72\) by dividing it by \(3\):
\[
72 = 3 \times 24
\]
3. Then, we take \(24\) and factor it further. \(24\) can be divided by \(2\):
\[
24 = 2 \times 12
\]
4. Next, we can factor \(12\) as:
\[
12 = 2 \times 6
\]
5. Finally, we have \(6\), which can be factored as:
\[
6 = 2 \times 3
\]
Now, we can fill in the factor tree:
```
72
/ \
3 24
/ \
2 12
/ \
2 6
/ \
2 3
```
So, the completed factor tree for \(72\) is as follows:
- **First level**: \(72 = 3 \times 24\)
- **Second level**: \(24 = 2 \times 12\)
- **Third level**: \(12 = 2 \times 6\)
- **Fourth level**: \(6 = 2 \times 3\)
Now, filling in the blanks in your tree with the whole numbers:
```
72
3•24
3•2•12
2•6
2•3
```
Thus, for your format:
- The first blank is \(24\).
- The second blank is \(2\).
- The third blank is \(12\).
- The fourth blank is \(6\).
- The fifth blank is \(3\).
Xpress 48 as a product of prime numbers using exponents
GPT-4o mini
To express \(48\) as a product of prime numbers using exponents, we can start by determining its prime factorization:
1. Divide \(48\) by \(2\) (the smallest prime number):
\[
48 \div 2 = 24
\]
2. Divide \(24\) by \(2\):
\[
24 \div 2 = 12
\]
3. Divide \(12\) by \(2\):
\[
12 \div 2 = 6
\]
4. Divide \(6\) by \(2\):
\[
6 \div 2 = 3
\]
5. Finally, \(3\) is prime and cannot be divided further.
Now, we can summarize the prime factors:
- The prime \(2\) is used \(4\) times.
- The prime \(3\) is used \(1\) time.
Thus, the prime factorization of \(48\) is:
\[
48 = 2^4 \cdot 3^1
\]
So, expressing \(48\) as a product of prime numbers using exponents gives us:
\[
2^4 \cdot 3^1
\]
1. Divide \(48\) by \(2\) (the smallest prime number):
\[
48 \div 2 = 24
\]
2. Divide \(24\) by \(2\):
\[
24 \div 2 = 12
\]
3. Divide \(12\) by \(2\):
\[
12 \div 2 = 6
\]
4. Divide \(6\) by \(2\):
\[
6 \div 2 = 3
\]
5. Finally, \(3\) is prime and cannot be divided further.
Now, we can summarize the prime factors:
- The prime \(2\) is used \(4\) times.
- The prime \(3\) is used \(1\) time.
Thus, the prime factorization of \(48\) is:
\[
48 = 2^4 \cdot 3^1
\]
So, expressing \(48\) as a product of prime numbers using exponents gives us:
\[
2^4 \cdot 3^1
\]