What type of number will result from 512⋅16−−√

Question

5
12
⋅
16
?(1 point)
Responses

Irrational, because one factor is irrational.
Irrational, because one factor is irrational.

Irrational, because both factors are irrational.
Irrational, because both factors are irrational.

Rational, because both numbers are rational.
Rational, because both numbers are rational.

Radical, because one factor is a radical.

Answer 1

To evaluate the expression \( 512 \cdot \sqrt{16} \):

1. Calculate \(\sqrt{16}\): \[ \sqrt{16} = 4 \]

2. Substitute \(\sqrt{16}\) back into the expression: \[ 512 \cdot \sqrt{16} = 512 \cdot 4 \]

3. Perform the multiplication: \[ 512 \cdot 4 = 2048 \]

Since both \( 512 \) and \( 4 \) are rational numbers, their product, \( 2048 \), is also a rational number.

Thus, the answer is: Rational, because both numbers are rational.

Answer 2

Find the product of 5–√
5
and 16
1
6
. What type of number is it?(1 point)
Responses

2.40; a rational number
2.40; a rational number

0.37; a rational number
0.37; a rational number

2.4027 . . . ; an irrational number
2.4027 . . . ; an irrational number

0.3726 . . . ; an irrational number

Answer 3

To find the product of \( 5 - \sqrt{5} \) and \( 16 \), we can calculate it as follows:

1. Calculate the product: \[ (5 - \sqrt{5}) \cdot 16 = 16 \cdot 5 - 16 \cdot \sqrt{5} \] \[ = 80 - 16\sqrt{5} \]

2. Determine the type of the number:

The term \( 80 \) is a rational number, and \( 16\sqrt{5} \) involves \( \sqrt{5} \), which is irrational. The expression \( 80 - 16\sqrt{5} \) combines a rational number and an irrational number, resulting in an irrational number.

Thus, the product \( 80 - 16\sqrt{5} \) is an irrational number.

Given the response options, none of the provided numerical answers (e.g., 2.40, 0.37) correspond to the exact value of \( 80 - 16\sqrt{5} \), but the important part of the question is identifying the type of number.

Since the expression leads to an irrational number, we conclude with:

Answer: \( 80 - 16\sqrt{5} \) ; an irrational number.

If required to select from the provided options, please choose one that states it is an irrational number. However, none seem to be correctly related to the calculation done here for that specific product.

Answer 4

What is the sum of the following expression?

2108−−−√+48−−√+527−−√
2
108
+
48
+
5
27

(1 point)
Responses

8183−−−√
8
183
8 square root of 183

31183−−−√
31
183
31 square root of 183

133–√
13
3
13 square root of 3

313–√

Answer 5

To find the sum of the expression \( \sqrt{2108} + \sqrt{48} + \sqrt{527} \), we first simplify each square root.

1. Simplifying \( \sqrt{2108} \):

   • Factor 2108 to find perfect squares: \[ 2108 = 4 \times 527; \quad \sqrt{2108} = \sqrt{4 \times 527} = \sqrt{4} \cdot \sqrt{527} = 2\sqrt{527} \]

2. Simplifying \( \sqrt{48} \):

   • Factor 48 to find perfect squares: \[ 48 = 16 \times 3; \quad \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3} \]

3. Simplifying \( \sqrt{527} \):

   • \( 527 \) cannot be simplified further since it is not a perfect square and doesn't have perfect square factors.

4. Combining the results:

   • Now we add: \[ \sqrt{2108} + \sqrt{48} + \sqrt{527} = 2\sqrt{527} + 4\sqrt{3} + \sqrt{527} \]
   • Combine like terms: \[ = (2\sqrt{527} + \sqrt{527}) + 4\sqrt{3} = 3\sqrt{527} + 4\sqrt{3} \]

Since we obtained \( 3\sqrt{527} + 4\sqrt{3} \), none of the options directly match this exact form, but we need to determine which choice closely resembles this.

Among the options given:

1. 8183−−−√ - not matched
2. 31183−−−√ - not matched
3. 133–√ - not matched
4. 313–√ - not matched

None of the provided options match the final result \( 3\sqrt{527} + 4\sqrt{3} \).

If there is a specific correct answer indicated by the question context that seems to be the closest or a common reduction approach, please double-check the options or review the computation steps related to \( \sqrt{2108}, \sqrt{48}, \sqrt{527} \) rational roots or combinations.

However, based on the correct addition and simplification, the sum is \( 3\sqrt{527} + 4\sqrt{3} \), which does not match any of the options provided.

Answer 6

you have to choose one

Answer 7

I apologize for the confusion earlier. Let's analyze the provided options in order to select the best one based on simplifications. Recall that we calculated:

\[ \sqrt{2108} + \sqrt{48} + \sqrt{527} = 3\sqrt{527} + 4\sqrt{3} \]

Now let's look at the provided options:

1. \( 8\sqrt{183} \)
2. \( 31\sqrt{183} \)
3. \( 13\sqrt{3} \)
4. \( 3\sqrt{13} \)

Examining Each Option:

1. \( 8\sqrt{183} \): This does not match our expression.
2. \( 31\sqrt{183} \): This does not match our expression.
3. \( 13\sqrt{3} \): This contains \( \sqrt{3} \) but does not account for \( 3\sqrt{527} \).
4. \( 3\sqrt{13} \): This form is not directly fitting our results.

Among these options, the one that involves a term with \( \sqrt{3} \) is \( 13\sqrt{3} \), which at least contains the \( \sqrt{3} \) component, even though \( 3\sqrt{527} \) is missing.

Conclusion:

Given the ambiguity and lack of complete match, the best choice to select from the list provided would be:

\( 13\sqrt{3} \).

This is the only option that includes a component similar to one of our results, even though it doesn’t reflect the complete sum.