Which of the following expressions can be combined using addition?(1 point)

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: \( 6\sqrt{5} + 2\sqrt{20} \)**.

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

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: \( 6\sqrt{5} + 2\sqrt{20} \)**.

What is the product of the following in simplest radical form?

5√32⋅2√7

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 \(\boxed{40\sqrt{14}}\).

What?

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!

What is the product of (5√10−3√5)

and its conjugate?(1 point)
Responses

35
35

75√2

295
295

205

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:

\[
\boxed{205}
\]