Question

What type of number will result from multiplying -√8 and ? 5
an irrational number
a rational number
the number zero
an integer

Answer 1

Simon, Asia, and Jiovanni are finding the product of a rational and irrational number.
Simon says the result should be irrational because the product of an irrational and rational number is always irrational.
Asia says the result should be zero because the product of an irrational and rational number is always zero.
Jiovanni says the result should be rational because the product of an irrational and rational number is always rational.
Who is correct?
(1 point)
None of them are correct.
Simon
Jiovanni
Asia

Answer 2

Which of the following expressions can be combined using addition?
6sqrt(5) + 2sqrt(20)
3sqrt(13) + 4sqrt(17)
3sqrt(28) + 2sqrt(44)
2sqrt(48) + 4sqrt(98)

Answer 3

Which of the following expressions can be combined using subtraction?
6sqrt(45) - 2sqrt(63)
5sqrt(150) - 3sqrt(32)
8sqrt(17) - 6sqrt(13)
8sqrt(99) - 2sqrt(176)

Answer 4

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

5sqrt(32) * 2sqrt(7)

Answer 5

What is the quotient of the following expression? State the answer in simplest radical form. Rationalize the denominator if necessary.
(12sqrt(24))/(3sqrt(300))

Answer 6

To determine what type of number results from multiplying \(-\sqrt{8}\) and \(5\), we can first simplify \(-\sqrt{8}\).

The square root of 8 can be simplified as follows:

\[
\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}
\]

Now, substituting this back, we have:

\[
-\sqrt{8} = -2\sqrt{2}
\]

Now, we can multiply this value by \(5\):

\[
-2\sqrt{2} \cdot 5 = -10\sqrt{2}
\]

The result, \(-10\sqrt{2}\), is irrational because it involves multiplying a rational number (\(-10\)) with an irrational number (\(\sqrt{2}\)).

Therefore, the answer to the question is:

**An irrational number.**

Answer 7

To evaluate who is correct among Simon, Asia, and Jiovanni, let's analyze their statements regarding the product of a rational and irrational number.

1. **Simon says:** The result should be irrational because the product of an irrational and rational number is always irrational.
- This statement is **correct**. The product of a non-zero rational number and an irrational number is indeed always irrational. (Note: If the rational number is zero, the product would be zero, which is a special case.)

2. **Asia says:** The result should be zero because the product of an irrational and rational number is always zero.
- This statement is **incorrect**. The product of an irrational number and a rational number is not always zero; it is only zero if the rational number is zero.

3. **Jiovanni says:** The result should be rational because the product of an irrational and rational number is always rational.
- This statement is **incorrect**. The product of an irrational number and a non-zero rational number is not rational.

Based on this analysis, the correct answer is:

**Simon.**

Answer 8

To determine which of the given expressions can be combined using addition, we need to check if the terms under the square roots can be simplified to have the same radical.

1. **\(6\sqrt{5} + 2\sqrt{20}\)**
- Simplifying \(2\sqrt{20}\):
\(\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}\)
So, \(2\sqrt{20} = 2 \times 2\sqrt{5} = 4\sqrt{5}\).
- Now we can rewrite the expression:
\(6\sqrt{5} + 4\sqrt{5} = (6 + 4)\sqrt{5} = 10\sqrt{5}\).
- This expression can be combined.

2. **\(3\sqrt{13} + 4\sqrt{17}\)**
- The radicals \(\sqrt{13}\) and \(\sqrt{17}\) are not the same.
- This expression cannot be combined.

3. **\(3\sqrt{28} + 2\sqrt{44}\)**
- Simplifying \(3\sqrt{28}\):
\(\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}\)
So, \(3\sqrt{28} = 3 \times 2\sqrt{7} = 6\sqrt{7}\).
- Simplifying \(2\sqrt{44}\):
\(\sqrt{44} = \sqrt{4 \times 11} = 2\sqrt{11}\)
So, \(2\sqrt{44} = 2 \times 2\sqrt{11} = 4\sqrt{11}\).
- The radicals \(\sqrt{7}\) and \(\sqrt{11}\) are not the same.
- This expression cannot be combined.

4. **\(2\sqrt{48} + 4\sqrt{98}\)**
- Simplifying \(2\sqrt{48}\):
\(\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}\)
So, \(2\sqrt{48} = 2 \times 4\sqrt{3} = 8\sqrt{3}\).
- Simplifying \(4\sqrt{98}\):
\(\sqrt{98} = \sqrt{49 \times 2} = 7\sqrt{2}\)
So, \(4\sqrt{98} = 4 \times 7\sqrt{2} = 28\sqrt{2}\).
- The radicals \(\sqrt{3}\) and \(\sqrt{2}\) are not the same.
- This expression cannot be combined.

**Conclusion:** The only expression that can be combined using addition is:

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

Answer 9

To determine which of the given expressions can be combined using subtraction, we need to check if the terms under the square roots can be simplified to have the same radical.

1. **\(6\sqrt{45} - 2\sqrt{63}\)**
- Simplifying \(6\sqrt{45}\):
\(\sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5}\)
So, \(6\sqrt{45} = 6 \times 3\sqrt{5} = 18\sqrt{5}\).
- Simplifying \(2\sqrt{63}\):
\(\sqrt{63} = \sqrt{9 \cdot 7} = 3\sqrt{7}\)
So, \(2\sqrt{63} = 2 \times 3\sqrt{7} = 6\sqrt{7}\).
- The radicals \(\sqrt{5}\) and \(\sqrt{7}\) are not the same.
- This expression cannot be combined.

2. **\(5\sqrt{150} - 3\sqrt{32}\)**
- Simplifying \(5\sqrt{150}\):
\(\sqrt{150} = \sqrt{25 \cdot 6} = 5\sqrt{6}\)
So, \(5\sqrt{150} = 5 \times 5\sqrt{6} = 25\sqrt{6}\).
- Simplifying \(3\sqrt{32}\):
\(\sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}\)
So, \(3\sqrt{32} = 3 \times 4\sqrt{2} = 12\sqrt{2}\).
- The radicals \(\sqrt{6}\) and \(\sqrt{2}\) are not the same.
- This expression cannot be combined.

3. **\(8\sqrt{17} - 6\sqrt{13}\)**
- The radicals \(\sqrt{17}\) and \(\sqrt{13}\) are not the same.
- This expression cannot be combined.

4. **\(8\sqrt{99} - 2\sqrt{176}\)**
- Simplifying \(8\sqrt{99}\):
\(\sqrt{99} = \sqrt{9 \cdot 11} = 3\sqrt{11}\)
So, \(8\sqrt{99} = 8 \times 3\sqrt{11} = 24\sqrt{11}\).
- Simplifying \(2\sqrt{176}\):
\(\sqrt{176} = \sqrt{16 \cdot 11} = 4\sqrt{11}\)
So, \(2\sqrt{176} = 2 \times 4\sqrt{11} = 8\sqrt{11}\).
- Both terms now have the same radical:
\(24\sqrt{11} - 8\sqrt{11} = (24 - 8)\sqrt{11} = 16\sqrt{11}\).
- This expression **can be combined**.

**Conclusion:** The expression that can be combined using subtraction is:

**\(8\sqrt{99} - 2\sqrt{176}\)**.

Answer 10

To find the product of the expression \(5\sqrt{32} \times 2\sqrt{7}\), follow these steps:

1. **Multiply the coefficients**:
\[
5 \times 2 = 10
\]

2. **Multiply the radicals**:
\[
\sqrt{32} \times \sqrt{7} = \sqrt{32 \times 7} = \sqrt{224}
\]

3. **Combine the results**:
\[
10\sqrt{224}
\]

4. **Simplify \(\sqrt{224}\)**:
- First, factor \(224\):
\[
224 = 16 \times 14 = 16 \times 2 \times 7
\]
- Now take the square root:
\[
\sqrt{224} = \sqrt{16 \times 14} = \sqrt{16} \times \sqrt{14} = 4\sqrt{14}
\]

5. **Substitute back**:
\[
10\sqrt{224} = 10 \times 4\sqrt{14} = 40\sqrt{14}
\]

Thus, the product of \(5\sqrt{32} \times 2\sqrt{7}\) in simplest radical form is

\[
\boxed{40\sqrt{14}}.
\]

Answer 11

To simplify the expression \(\frac{12\sqrt{24}}{3\sqrt{300}}\), follow these steps:

1. **Simplify the coefficients**:
\[
\frac{12}{3} = 4
\]
So, we have:
\[
\frac{12\sqrt{24}}{3\sqrt{300}} = 4 \cdot \frac{\sqrt{24}}{\sqrt{300}}
\]

2. **Simplify the radicals**:
We can combine the radicals:
\[
\frac{\sqrt{24}}{\sqrt{300}} = \sqrt{\frac{24}{300}}
\]

3. **Simplify \(\frac{24}{300}\)**:
\[
\frac{24}{300} = \frac{24 \div 12}{300 \div 12} = \frac{2}{25}
\]
So, we now have:
\[
\sqrt{\frac{24}{300}} = \sqrt{\frac{2}{25}} = \frac{\sqrt{2}}{\sqrt{25}} = \frac{\sqrt{2}}{5}
\]

4. **Combine everything**:
Now, substitute back:
\[
4 \cdot \frac{\sqrt{24}}{\sqrt{300}} = 4 \cdot \frac{\sqrt{2}}{5} = \frac{4\sqrt{2}}{5}
\]

The final result in simplest radical form is:
\[
\boxed{\frac{4\sqrt{2}}{5}}
\]