Asked by Bot GPT 4-o mini
Which of the following expressions can be combined using addition?(1 point)
Responses
65–√+220−−√
6 square root of 5 plus 2 square root of 20
248−−√+498−−√
2 square root of 48 plus 4 square root of 98
328−−√+244−−√
3 square root of 28 plus 2 square root of 44
313−−√+417−−√
Responses
65–√+220−−√
6 square root of 5 plus 2 square root of 20
248−−√+498−−√
2 square root of 48 plus 4 square root of 98
328−−√+244−−√
3 square root of 28 plus 2 square root of 44
313−−√+417−−√
Answers
Answered by
Bot GPT 4-o mini
Which of the following expressions can be combined using addition?(1 point)
Responses
6√5+2√20
2√48+4√98
3√28+2√44
3√13+4√17
Responses
6√5+2√20
2√48+4√98
3√28+2√44
3√13+4√17
Answered by
Bot GPT 4-o mini
What is the product of the following in simplest radical form?
5√32⋅2√7
5√32⋅2√7
Answered by
Bot GPT 4-o mini
What?
Answered by
Bot GPT 4-o mini
What is the product of (5√10−3√5)
and its conjugate?(1 point)
Responses
35
35
75√2
295
295
205
and its conjugate?(1 point)
Responses
35
35
75√2
295
295
205
Answered by
GPT-4o mini
To determine which of the given expressions can be combined using addition, we need to simplify the square root expressions and see if they result in similar terms.
1. **Expression 1: \( 65\sqrt{} + 220\sqrt{} \)**
- This notation seems unclear. If it refers to \( 65\sqrt{a} + 220\sqrt{b} \), they can't be combined without knowing values for \(a\) and \(b\).
2. **Expression 2: \( 6\sqrt{5} + 2\sqrt{20} \)**
- Simplifying \( \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} \)
- Thus, \( 2\sqrt{20} = 2(2\sqrt{5}) = 4\sqrt{5} \)
- Therefore, \( 6\sqrt{5} + 4\sqrt{5} = 10\sqrt{5} \) (This expression can be combined.)
3. **Expression 3: \( 248\sqrt{} + 498\sqrt{} \)**
- Similar issue as Expression 1. Not clear notation. Cannot be combined without clarity.
4. **Expression 4: \( 3\sqrt{28} + 2\sqrt{44} \)**
- Simplifying \( \sqrt{28} = \sqrt{4 \cdot 7} = 2\sqrt{7} \)
- Thus, \( 3\sqrt{28} = 3(2\sqrt{7}) = 6\sqrt{7} \)
- Now simplifying \( \sqrt{44} = \sqrt{4 \cdot 11} = 2\sqrt{11} \)
- Thus, \( 2\sqrt{44} = 2(2\sqrt{11}) = 4\sqrt{11} \)
- Since \( 6\sqrt{7} + 4\sqrt{11} \) can't be combined, this expression cannot be combined.
5. **Expression 5: \( 3\sqrt{13} + 4\sqrt{17} \)**
- The terms \( \sqrt{13} \) and \( \sqrt{17} \) are different roots, so they cannot be combined.
In summary, the only expression that can be combined using addition is **Expression 2: \( 6\sqrt{5} + 2\sqrt{20} \)**.
1. **Expression 1: \( 65\sqrt{} + 220\sqrt{} \)**
- This notation seems unclear. If it refers to \( 65\sqrt{a} + 220\sqrt{b} \), they can't be combined without knowing values for \(a\) and \(b\).
2. **Expression 2: \( 6\sqrt{5} + 2\sqrt{20} \)**
- Simplifying \( \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} \)
- Thus, \( 2\sqrt{20} = 2(2\sqrt{5}) = 4\sqrt{5} \)
- Therefore, \( 6\sqrt{5} + 4\sqrt{5} = 10\sqrt{5} \) (This expression can be combined.)
3. **Expression 3: \( 248\sqrt{} + 498\sqrt{} \)**
- Similar issue as Expression 1. Not clear notation. Cannot be combined without clarity.
4. **Expression 4: \( 3\sqrt{28} + 2\sqrt{44} \)**
- Simplifying \( \sqrt{28} = \sqrt{4 \cdot 7} = 2\sqrt{7} \)
- Thus, \( 3\sqrt{28} = 3(2\sqrt{7}) = 6\sqrt{7} \)
- Now simplifying \( \sqrt{44} = \sqrt{4 \cdot 11} = 2\sqrt{11} \)
- Thus, \( 2\sqrt{44} = 2(2\sqrt{11}) = 4\sqrt{11} \)
- Since \( 6\sqrt{7} + 4\sqrt{11} \) can't be combined, this expression cannot be combined.
5. **Expression 5: \( 3\sqrt{13} + 4\sqrt{17} \)**
- The terms \( \sqrt{13} \) and \( \sqrt{17} \) are different roots, so they cannot be combined.
In summary, the only expression that can be combined using addition is **Expression 2: \( 6\sqrt{5} + 2\sqrt{20} \)**.
Answered by
GPT-4o mini
To determine which of the given expressions can be combined using addition, we will simplify each expression and check if they contain like terms (i.e., terms that have the same base under the square root) so they can be added together.
1. **Expression 1: \( 6\sqrt{5} + 2\sqrt{20} \)**
- Simplify \( \sqrt{20} \):
\[
\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}
\]
- So, \( 2\sqrt{20} = 2(2\sqrt{5}) = 4\sqrt{5} \).
- Now combine:
\[
6\sqrt{5} + 4\sqrt{5} = 10\sqrt{5}
\]
- **This expression can be combined.**
2. **Expression 2: \( 2\sqrt{48} + 4\sqrt{98} \)**
- Simplify \( \sqrt{48} \):
\[
\sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3}
\]
\[
2\sqrt{48} = 2(4\sqrt{3}) = 8\sqrt{3}
\]
- Simplify \( \sqrt{98} \):
\[
\sqrt{98} = \sqrt{49 \cdot 2} = 7\sqrt{2}
\]
\[
4\sqrt{98} = 4(7\sqrt{2}) = 28\sqrt{2}
\]
- Now we have:
\[
8\sqrt{3} + 28\sqrt{2}
\]
- Since \( \sqrt{3} \) and \( \sqrt{2} \) are different radical terms, **this expression cannot be combined.**
3. **Expression 3: \( 3\sqrt{28} + 2\sqrt{44} \)**
- Simplify \( \sqrt{28} \):
\[
\sqrt{28} = \sqrt{4 \cdot 7} = 2\sqrt{7}
\]
\[
3\sqrt{28} = 3(2\sqrt{7}) = 6\sqrt{7}
\]
- Simplify \( \sqrt{44} \):
\[
\sqrt{44} = \sqrt{4 \cdot 11} = 2\sqrt{11}
\]
\[
2\sqrt{44} = 2(2\sqrt{11}) = 4\sqrt{11}
\]
- Now we have:
\[
6\sqrt{7} + 4\sqrt{11}
\]
- Since \( \sqrt{7} \) and \( \sqrt{11} \) are different radical terms, **this expression cannot be combined.**
4. **Expression 4: \( 3\sqrt{13} + 4\sqrt{17} \)**
- Neither \( \sqrt{13} \) nor \( \sqrt{17} \) can be simplified further, and they are different. Thus, **this expression cannot be combined.**
In conclusion:
- The only expression that can be combined is **Expression 1: \( 6\sqrt{5} + 2\sqrt{20} \)**.
1. **Expression 1: \( 6\sqrt{5} + 2\sqrt{20} \)**
- Simplify \( \sqrt{20} \):
\[
\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}
\]
- So, \( 2\sqrt{20} = 2(2\sqrt{5}) = 4\sqrt{5} \).
- Now combine:
\[
6\sqrt{5} + 4\sqrt{5} = 10\sqrt{5}
\]
- **This expression can be combined.**
2. **Expression 2: \( 2\sqrt{48} + 4\sqrt{98} \)**
- Simplify \( \sqrt{48} \):
\[
\sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3}
\]
\[
2\sqrt{48} = 2(4\sqrt{3}) = 8\sqrt{3}
\]
- Simplify \( \sqrt{98} \):
\[
\sqrt{98} = \sqrt{49 \cdot 2} = 7\sqrt{2}
\]
\[
4\sqrt{98} = 4(7\sqrt{2}) = 28\sqrt{2}
\]
- Now we have:
\[
8\sqrt{3} + 28\sqrt{2}
\]
- Since \( \sqrt{3} \) and \( \sqrt{2} \) are different radical terms, **this expression cannot be combined.**
3. **Expression 3: \( 3\sqrt{28} + 2\sqrt{44} \)**
- Simplify \( \sqrt{28} \):
\[
\sqrt{28} = \sqrt{4 \cdot 7} = 2\sqrt{7}
\]
\[
3\sqrt{28} = 3(2\sqrt{7}) = 6\sqrt{7}
\]
- Simplify \( \sqrt{44} \):
\[
\sqrt{44} = \sqrt{4 \cdot 11} = 2\sqrt{11}
\]
\[
2\sqrt{44} = 2(2\sqrt{11}) = 4\sqrt{11}
\]
- Now we have:
\[
6\sqrt{7} + 4\sqrt{11}
\]
- Since \( \sqrt{7} \) and \( \sqrt{11} \) are different radical terms, **this expression cannot be combined.**
4. **Expression 4: \( 3\sqrt{13} + 4\sqrt{17} \)**
- Neither \( \sqrt{13} \) nor \( \sqrt{17} \) can be simplified further, and they are different. Thus, **this expression cannot be combined.**
In conclusion:
- The only expression that can be combined is **Expression 1: \( 6\sqrt{5} + 2\sqrt{20} \)**.
Answered by
GPT-4o mini
To find the product of \(5\sqrt{32} \cdot 2\sqrt{7}\), we can use the properties of multiplication of radicals.
1. **Multiply the coefficients (the numbers in front of the radicals)**:
\[
5 \cdot 2 = 10
\]
2. **Multiply the radicals**:
\[
\sqrt{32} \cdot \sqrt{7} = \sqrt{32 \cdot 7} = \sqrt{224}
\]
3. **Now, combine the results**:
\[
5\sqrt{32} \cdot 2\sqrt{7} = 10\sqrt{224}
\]
4. **Next, simplify \(\sqrt{224}\)**:
- First, factor \(224\):
\[
224 = 16 \cdot 14 = 16 \cdot 2 \cdot 7
\]
- Taking the square root:
\[
\sqrt{224} = \sqrt{16 \cdot 14} = \sqrt{16} \cdot \sqrt{14} = 4\sqrt{14}
\]
5. **Substituting back to get the final product**:
\[
10\sqrt{224} = 10 \cdot 4\sqrt{14} = 40\sqrt{14}
\]
Thus, the product of \(5\sqrt{32} \cdot 2\sqrt{7}\) in simplest radical form is \(\boxed{40\sqrt{14}}\).
1. **Multiply the coefficients (the numbers in front of the radicals)**:
\[
5 \cdot 2 = 10
\]
2. **Multiply the radicals**:
\[
\sqrt{32} \cdot \sqrt{7} = \sqrt{32 \cdot 7} = \sqrt{224}
\]
3. **Now, combine the results**:
\[
5\sqrt{32} \cdot 2\sqrt{7} = 10\sqrt{224}
\]
4. **Next, simplify \(\sqrt{224}\)**:
- First, factor \(224\):
\[
224 = 16 \cdot 14 = 16 \cdot 2 \cdot 7
\]
- Taking the square root:
\[
\sqrt{224} = \sqrt{16 \cdot 14} = \sqrt{16} \cdot \sqrt{14} = 4\sqrt{14}
\]
5. **Substituting back to get the final product**:
\[
10\sqrt{224} = 10 \cdot 4\sqrt{14} = 40\sqrt{14}
\]
Thus, the product of \(5\sqrt{32} \cdot 2\sqrt{7}\) in simplest radical form is \(\boxed{40\sqrt{14}}\).
Answered by
GPT-4o mini
I apologize if my previous explanation was confusing. Let's go through the calculation step by step clearly.
We want to calculate the product:
\[
5\sqrt{32} \cdot 2\sqrt{7}
\]
### Step 1: Multiply the Coefficients
First, we multiply the coefficients (the numbers in front of the square roots):
\[
5 \cdot 2 = 10
\]
### Step 2: Multiply the Radicals
Next, we multiply the square roots:
\[
\sqrt{32} \cdot \sqrt{7} = \sqrt{32 \cdot 7} = \sqrt{224}
\]
### Step 3: Combine the Results
Now we combine our results from Steps 1 and 2:
\[
5\sqrt{32} \cdot 2\sqrt{7} = 10\sqrt{224}
\]
### Step 4: Simplify \(\sqrt{224}\)
Now we simplify \(\sqrt{224}\):
1. Factor \(224\):
\[
224 = 16 \cdot 14
\]
2. Take the square root:
\[
\sqrt{224} = \sqrt{16 \cdot 14} = \sqrt{16} \cdot \sqrt{14} = 4\sqrt{14}
\]
### Final Step: Substitute Back
Now substitute back into our expression:
\[
10\sqrt{224} = 10 \cdot 4\sqrt{14} = 40\sqrt{14}
\]
Thus, the final answer is:
\[
\boxed{40\sqrt{14}}
\]
If there are any clarifications or specific concerns you'd like to address, please let me know!
We want to calculate the product:
\[
5\sqrt{32} \cdot 2\sqrt{7}
\]
### Step 1: Multiply the Coefficients
First, we multiply the coefficients (the numbers in front of the square roots):
\[
5 \cdot 2 = 10
\]
### Step 2: Multiply the Radicals
Next, we multiply the square roots:
\[
\sqrt{32} \cdot \sqrt{7} = \sqrt{32 \cdot 7} = \sqrt{224}
\]
### Step 3: Combine the Results
Now we combine our results from Steps 1 and 2:
\[
5\sqrt{32} \cdot 2\sqrt{7} = 10\sqrt{224}
\]
### Step 4: Simplify \(\sqrt{224}\)
Now we simplify \(\sqrt{224}\):
1. Factor \(224\):
\[
224 = 16 \cdot 14
\]
2. Take the square root:
\[
\sqrt{224} = \sqrt{16 \cdot 14} = \sqrt{16} \cdot \sqrt{14} = 4\sqrt{14}
\]
### Final Step: Substitute Back
Now substitute back into our expression:
\[
10\sqrt{224} = 10 \cdot 4\sqrt{14} = 40\sqrt{14}
\]
Thus, the final answer is:
\[
\boxed{40\sqrt{14}}
\]
If there are any clarifications or specific concerns you'd like to address, please let me know!
Answered by
GPT-4o mini
To find the product of \( (5\sqrt{10} - 3\sqrt{5}) \) and its conjugate \( (5\sqrt{10} + 3\sqrt{5}) \), we can use the difference of squares formula, which states:
\[
(a - b)(a + b) = a^2 - b^2
\]
### Step 1: Identify \(a\) and \(b\)
Here:
- \( a = 5\sqrt{10} \)
- \( b = 3\sqrt{5} \)
### Step 2: Calculate \(a^2\) and \(b^2\)
1. **Calculate \(a^2\)**:
\[
a^2 = (5\sqrt{10})^2 = 5^2 \cdot (\sqrt{10})^2 = 25 \cdot 10 = 250
\]
2. **Calculate \(b^2\)**:
\[
b^2 = (3\sqrt{5})^2 = 3^2 \cdot (\sqrt{5})^2 = 9 \cdot 5 = 45
\]
### Step 3: Use the Difference of Squares
Now, plug in the values into the difference of squares formula:
\[
(5\sqrt{10} - 3\sqrt{5})(5\sqrt{10} + 3\sqrt{5}) = a^2 - b^2 = 250 - 45 = 205
\]
### Final Answer
The product is:
\[
\boxed{205}
\]
\[
(a - b)(a + b) = a^2 - b^2
\]
### Step 1: Identify \(a\) and \(b\)
Here:
- \( a = 5\sqrt{10} \)
- \( b = 3\sqrt{5} \)
### Step 2: Calculate \(a^2\) and \(b^2\)
1. **Calculate \(a^2\)**:
\[
a^2 = (5\sqrt{10})^2 = 5^2 \cdot (\sqrt{10})^2 = 25 \cdot 10 = 250
\]
2. **Calculate \(b^2\)**:
\[
b^2 = (3\sqrt{5})^2 = 3^2 \cdot (\sqrt{5})^2 = 9 \cdot 5 = 45
\]
### Step 3: Use the Difference of Squares
Now, plug in the values into the difference of squares formula:
\[
(5\sqrt{10} - 3\sqrt{5})(5\sqrt{10} + 3\sqrt{5}) = a^2 - b^2 = 250 - 45 = 205
\]
### Final Answer
The product is:
\[
\boxed{205}
\]
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