Asked by x
What is the sum of 35–√ +520−−√
?
?
Answers
Answered by
GPT-4o mini
To find the sum of \( \sqrt{35} + \sqrt{520} \), we first simplify \(\sqrt{520}\).
1. Factor \(520\):
\[
520 = 2^3 \times 5 \times 13
\]
2. Rewrite \(\sqrt{520}\):
\[
\sqrt{520} = \sqrt{2^3 \times 5 \times 13} = \sqrt{2^2 \times 2 \times 5 \times 13} = \sqrt{4 \times 2 \times 5 \times 13}
\]
\[
= \sqrt{4} \times \sqrt{2 \times 5 \times 13} = 2\sqrt{130}
\]
Now we can express the original sum:
\[
\sqrt{35} + \sqrt{520} = \sqrt{35} + 2\sqrt{130}
\]
This is the simplified form of the expression. Therefore, the sum is:
\[
\sqrt{35} + 2\sqrt{130}
\]
1. Factor \(520\):
\[
520 = 2^3 \times 5 \times 13
\]
2. Rewrite \(\sqrt{520}\):
\[
\sqrt{520} = \sqrt{2^3 \times 5 \times 13} = \sqrt{2^2 \times 2 \times 5 \times 13} = \sqrt{4 \times 2 \times 5 \times 13}
\]
\[
= \sqrt{4} \times \sqrt{2 \times 5 \times 13} = 2\sqrt{130}
\]
Now we can express the original sum:
\[
\sqrt{35} + \sqrt{520} = \sqrt{35} + 2\sqrt{130}
\]
This is the simplified form of the expression. Therefore, the sum is:
\[
\sqrt{35} + 2\sqrt{130}
\]
Answered by
GPT-4o mini
Let's go through the calculations step by step to ensure accuracy.
You want to find the sum of \( \sqrt{35} + \sqrt{520} \).
1. **Calculate \(\sqrt{520}\)**:
- First, we can simplify \(\sqrt{520}\):
\[
520 = 4 \times 130
\]
\[
\sqrt{520} = \sqrt{4 \times 130} = \sqrt{4} \times \sqrt{130} = 2\sqrt{130}
\]
2. **Write the sum**:
Now substitute back into the original expression:
\[
\sqrt{35} + \sqrt{520} = \sqrt{35} + 2\sqrt{130}
\]
So the final answer remains:
\[
\sqrt{35} + 2\sqrt{130}
\]
If you're looking for a numerical approximation:
- \(\sqrt{35} \approx 5.916\)
- \(\sqrt{130} \approx 11.401\), thus \(2\sqrt{130} \approx 22.802\)
Adding those approximate values:
\[
\sqrt{35} + 2\sqrt{130} \approx 5.916 + 22.802 \approx 28.718
\]
So, the expression in its simplest form is \( \sqrt{35} + 2\sqrt{130} \), and its numerical approximation is about \(28.718\). Let me know if you need further assistance!
You want to find the sum of \( \sqrt{35} + \sqrt{520} \).
1. **Calculate \(\sqrt{520}\)**:
- First, we can simplify \(\sqrt{520}\):
\[
520 = 4 \times 130
\]
\[
\sqrt{520} = \sqrt{4 \times 130} = \sqrt{4} \times \sqrt{130} = 2\sqrt{130}
\]
2. **Write the sum**:
Now substitute back into the original expression:
\[
\sqrt{35} + \sqrt{520} = \sqrt{35} + 2\sqrt{130}
\]
So the final answer remains:
\[
\sqrt{35} + 2\sqrt{130}
\]
If you're looking for a numerical approximation:
- \(\sqrt{35} \approx 5.916\)
- \(\sqrt{130} \approx 11.401\), thus \(2\sqrt{130} \approx 22.802\)
Adding those approximate values:
\[
\sqrt{35} + 2\sqrt{130} \approx 5.916 + 22.802 \approx 28.718
\]
So, the expression in its simplest form is \( \sqrt{35} + 2\sqrt{130} \), and its numerical approximation is about \(28.718\). Let me know if you need further assistance!
Answered by
GPT-4o mini
To find the difference \( \sqrt{147} - \sqrt{57} \), we first simplify each square root if possible.
1. **Simplify \( \sqrt{147} \)**:
\[
147 = 49 \times 3 = 7^2 \times 3
\]
Thus,
\[
\sqrt{147} = \sqrt{49 \times 3} = \sqrt{49} \times \sqrt{3} = 7\sqrt{3}
\]
2. **Simplify \( \sqrt{57} \)**:
The number 57 does not have any perfect square factors other than 1:
\[
57 = 3 \times 19
\]
Therefore,
\[
\sqrt{57} \text{ cannot be simplified further.}
\]
Now, we put it all together:
\[
\sqrt{147} - \sqrt{57} = 7\sqrt{3} - \sqrt{57}
\]
This is the simplified form of the expression. Therefore, the answer is:
\[
7\sqrt{3} - \sqrt{57}
\]
If you need a numerical approximation:
- \(\sqrt{3} \approx 1.732\) so \(7\sqrt{3} \approx 7 \times 1.732 \approx 12.124\)
- \(\sqrt{57} \approx 7.550\)
Thus the difference approximately is:
\[
12.124 - 7.550 \approx 4.574
\]
So, \(7\sqrt{3} - \sqrt{57}\) is the exact answer, and its approximate numerical value is \(4.574\). Let me know if you need anything else!
1. **Simplify \( \sqrt{147} \)**:
\[
147 = 49 \times 3 = 7^2 \times 3
\]
Thus,
\[
\sqrt{147} = \sqrt{49 \times 3} = \sqrt{49} \times \sqrt{3} = 7\sqrt{3}
\]
2. **Simplify \( \sqrt{57} \)**:
The number 57 does not have any perfect square factors other than 1:
\[
57 = 3 \times 19
\]
Therefore,
\[
\sqrt{57} \text{ cannot be simplified further.}
\]
Now, we put it all together:
\[
\sqrt{147} - \sqrt{57} = 7\sqrt{3} - \sqrt{57}
\]
This is the simplified form of the expression. Therefore, the answer is:
\[
7\sqrt{3} - \sqrt{57}
\]
If you need a numerical approximation:
- \(\sqrt{3} \approx 1.732\) so \(7\sqrt{3} \approx 7 \times 1.732 \approx 12.124\)
- \(\sqrt{57} \approx 7.550\)
Thus the difference approximately is:
\[
12.124 - 7.550 \approx 4.574
\]
So, \(7\sqrt{3} - \sqrt{57}\) is the exact answer, and its approximate numerical value is \(4.574\). Let me know if you need anything else!
Answered by
GPT-4o mini
Let’s correctly express the difference \( \sqrt{147} - \sqrt{57} \) as a single square root, if possible.
### Calculating \( \sqrt{147} - \sqrt{57} \)
1. **Use the property of square roots**:
We know that \( a - b = \frac{(a^2 - b^2)}{(a + b)} \) can help us combine square roots.
Let:
\( a = \sqrt{147} \)
\( b = \sqrt{57} \)
Therefore:
\[
\sqrt{147} - \sqrt{57} = \frac{(147 - 57)}{(\sqrt{147} + \sqrt{57})}
\]
2. **Calculate \( 147 - 57 \)**:
\[
147 - 57 = 90
\]
3. **Now write down the expression**:
So we have:
\[
\sqrt{147} - \sqrt{57} = \frac{90}{\sqrt{147} + \sqrt{57}}
\]
However, this does not yield a single square root. Instead, to express the difference in a single square root form would not be straightforward, as neither \( \sqrt{147} \) nor \( \sqrt{57} \) can combine algebraically into a simple square root expression like that.
### Summary
Thus, the most straightforward expression for the difference \( \sqrt{147} - \sqrt{57} \) still remains:
\[
\sqrt{147} - \sqrt{57}
\]
If we need to express it in a different way, we can write it as:
\[
\sqrt{147} - \sqrt{57} = 7\sqrt{3} - \sqrt{57}
\]
### Exact Answer
So the simplest form as a difference of square roots is:
\[
\sqrt{147} - \sqrt{57}
\]
If exact numerical relations are required, it remains in this form, as combining them further into a single square root isn’t manageable without specific mathematical manipulation or context.
If you’re looking for a different format (like a common square root), please clarify, and I can assist further!
### Calculating \( \sqrt{147} - \sqrt{57} \)
1. **Use the property of square roots**:
We know that \( a - b = \frac{(a^2 - b^2)}{(a + b)} \) can help us combine square roots.
Let:
\( a = \sqrt{147} \)
\( b = \sqrt{57} \)
Therefore:
\[
\sqrt{147} - \sqrt{57} = \frac{(147 - 57)}{(\sqrt{147} + \sqrt{57})}
\]
2. **Calculate \( 147 - 57 \)**:
\[
147 - 57 = 90
\]
3. **Now write down the expression**:
So we have:
\[
\sqrt{147} - \sqrt{57} = \frac{90}{\sqrt{147} + \sqrt{57}}
\]
However, this does not yield a single square root. Instead, to express the difference in a single square root form would not be straightforward, as neither \( \sqrt{147} \) nor \( \sqrt{57} \) can combine algebraically into a simple square root expression like that.
### Summary
Thus, the most straightforward expression for the difference \( \sqrt{147} - \sqrt{57} \) still remains:
\[
\sqrt{147} - \sqrt{57}
\]
If we need to express it in a different way, we can write it as:
\[
\sqrt{147} - \sqrt{57} = 7\sqrt{3} - \sqrt{57}
\]
### Exact Answer
So the simplest form as a difference of square roots is:
\[
\sqrt{147} - \sqrt{57}
\]
If exact numerical relations are required, it remains in this form, as combining them further into a single square root isn’t manageable without specific mathematical manipulation or context.
If you’re looking for a different format (like a common square root), please clarify, and I can assist further!
Answered by
GPT-4o mini
To find the difference \( \sqrt{663} - \sqrt{428} \), we can start by simplifying both square roots if possible.
### Step 1: Simplify \( \sqrt{663} \)
The prime factorization of \( 663 \) is:
\[
663 = 3 \times 221 = 3 \times 13 \times 17
\]
Since there are no perfect square factors, \( \sqrt{663} \) cannot be simplified further.
### Step 2: Simplify \( \sqrt{428} \)
The prime factorization of \( 428 \) is:
\[
428 = 4 \times 107 = 2^2 \times 107
\]
Thus,
\[
\sqrt{428} = \sqrt{4 \times 107} = \sqrt{4} \times \sqrt{107} = 2\sqrt{107}
\]
### Step 3: Write the difference
Now, we can express the difference:
\[
\sqrt{663} - \sqrt{428} = \sqrt{663} - 2\sqrt{107}
\]
### Conclusion
So, the difference \( \sqrt{663} - \sqrt{428} \) can be written as:
\[
\sqrt{663} - 2\sqrt{107}
\]
If you need a numerical approximation:
- \(\sqrt{663} \approx 25.77\)
- \(\sqrt{428} \approx 20.70\), thus \(2\sqrt{107} \approx 2 \times 10.34 \approx 20.68\)
Adding these approximate values will give:
\[
25.77 - 20.68 \approx 5.09
\]
Thus the exact answer is:
\[
\sqrt{663} - 2\sqrt{107}
\]
And the approximate numerical value is about \(5.09\). Let me know if you have any other questions!
### Step 1: Simplify \( \sqrt{663} \)
The prime factorization of \( 663 \) is:
\[
663 = 3 \times 221 = 3 \times 13 \times 17
\]
Since there are no perfect square factors, \( \sqrt{663} \) cannot be simplified further.
### Step 2: Simplify \( \sqrt{428} \)
The prime factorization of \( 428 \) is:
\[
428 = 4 \times 107 = 2^2 \times 107
\]
Thus,
\[
\sqrt{428} = \sqrt{4 \times 107} = \sqrt{4} \times \sqrt{107} = 2\sqrt{107}
\]
### Step 3: Write the difference
Now, we can express the difference:
\[
\sqrt{663} - \sqrt{428} = \sqrt{663} - 2\sqrt{107}
\]
### Conclusion
So, the difference \( \sqrt{663} - \sqrt{428} \) can be written as:
\[
\sqrt{663} - 2\sqrt{107}
\]
If you need a numerical approximation:
- \(\sqrt{663} \approx 25.77\)
- \(\sqrt{428} \approx 20.70\), thus \(2\sqrt{107} \approx 2 \times 10.34 \approx 20.68\)
Adding these approximate values will give:
\[
25.77 - 20.68 \approx 5.09
\]
Thus the exact answer is:
\[
\sqrt{663} - 2\sqrt{107}
\]
And the approximate numerical value is about \(5.09\). Let me know if you have any other questions!
Answered by
GPT-4o mini
To find the expression \( \sqrt{548} - \sqrt{63} + \sqrt{212} \), we can simplify each square root if possible.
### Step 1: Simplify \( \sqrt{548} \)
The prime factorization of \( 548 \) is:
\[
548 = 4 \times 137 = 2^2 \times 137
\]
Thus,
\[
\sqrt{548} = \sqrt{4 \times 137} = \sqrt{4} \times \sqrt{137} = 2\sqrt{137}
\]
### Step 2: Simplify \( \sqrt{63} \)
The prime factorization of \( 63 \) is:
\[
63 = 9 \times 7 = 3^2 \times 7
\]
Thus,
\[
\sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} = 3\sqrt{7}
\]
### Step 3: Simplify \( \sqrt{212} \)
The prime factorization of \( 212 \) is:
\[
212 = 4 \times 53 = 2^2 \times 53
\]
Thus,
\[
\sqrt{212} = \sqrt{4 \times 53} = \sqrt{4} \times \sqrt{53} = 2\sqrt{53}
\]
### Step 4: Combine the Expression
Now substituting back into the original expression:
\[
\sqrt{548} - \sqrt{63} + \sqrt{212} = 2\sqrt{137} - 3\sqrt{7} + 2\sqrt{53}
\]
### Conclusion
So the final expression in simplified form is:
\[
2\sqrt{137} - 3\sqrt{7} + 2\sqrt{53}
\]
This is your final answer. If you have any further questions or need numerical approximations, please let me know!
### Step 1: Simplify \( \sqrt{548} \)
The prime factorization of \( 548 \) is:
\[
548 = 4 \times 137 = 2^2 \times 137
\]
Thus,
\[
\sqrt{548} = \sqrt{4 \times 137} = \sqrt{4} \times \sqrt{137} = 2\sqrt{137}
\]
### Step 2: Simplify \( \sqrt{63} \)
The prime factorization of \( 63 \) is:
\[
63 = 9 \times 7 = 3^2 \times 7
\]
Thus,
\[
\sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} = 3\sqrt{7}
\]
### Step 3: Simplify \( \sqrt{212} \)
The prime factorization of \( 212 \) is:
\[
212 = 4 \times 53 = 2^2 \times 53
\]
Thus,
\[
\sqrt{212} = \sqrt{4 \times 53} = \sqrt{4} \times \sqrt{53} = 2\sqrt{53}
\]
### Step 4: Combine the Expression
Now substituting back into the original expression:
\[
\sqrt{548} - \sqrt{63} + \sqrt{212} = 2\sqrt{137} - 3\sqrt{7} + 2\sqrt{53}
\]
### Conclusion
So the final expression in simplified form is:
\[
2\sqrt{137} - 3\sqrt{7} + 2\sqrt{53}
\]
This is your final answer. If you have any further questions or need numerical approximations, please let me know!