Question

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

Rational, because both numbers are rational.
Rational, because both numbers are rational. - not selected, this is the correct answer Why is this correct?

Irrational, because both factors are irrational.
Irrational, because both factors are irrational. - no response given

Radical, because one factor is a radical.
Radical, because one factor is a radical. - no response given

Irrational, because one factor is irrational.
Irrational, because one factor is irrational. - incorrect why is this incorrect

Answer 1

a little shorter

Answer 2

Question
What type of number will result from 5/12⋅16−−√
?
Responses

Rational, because both numbers are rational.
Rational, because both numbers are rational. - not selected, this is the correct answer why is this the correct answer?

Irrational, because both factors are irrational.
Irrational, because both factors are irrational. - no response given

Radical, because one factor is a radical.
Radical, because one factor is a radical. - no response given

Irrational, because one factor is irrational. why is this wrong

Answer 3

To determine the type of number that results from the expression \( 512 \cdot \sqrt{16} \), let's analyze both components of the expression:

1. **Evaluating \( \sqrt{16} \)**:
\[
\sqrt{16} = 4
\]
Since \( 4 \) is a rational number, we now have:
\[
512 \cdot 4
\]

2. **Evaluating \( 512 \cdot 4 \)**:
\[
512 \cdot 4 = 2048
\]
Since \( 2048 \) is also a rational number, the overall result is rational.

### Explanation of the Responses:

- **"Rational, because both numbers are rational."**: This response is correct because both \( 512 \) and \( \sqrt{16} \) (which equals \( 4 \)) are rational numbers, and the product of two rational numbers is always rational.

- **"Irrational, because both factors are irrational."**: This response is incorrect because \( \sqrt{16} = 4 \) is not an irrational number; it is a rational number.

- **"Radical, because one factor is a radical."**: The term "radical" is sometimes used informally, but it's not a classification of the number itself. In this case, while one factor is indeed a radical (\( \sqrt{16} \)), the result of the entire operation is still a rational number, so this sounds inaccurate.

- **"Irrational, because one factor is irrational."**: This response is incorrect and could be misleading because the factor \( \sqrt{16} \) is not irrational. Thus, the presence of an irrational factor does not apply here.

### Summary:
The correct classification is **rational** because both initial numbers are rational, and their product results in a rational number.

Answer 4

To determine the type of number from \( 512 \cdot \sqrt{16} \):

1. **Calculate \( \sqrt{16} \)**:
\[
\sqrt{16} = 4 \quad \text{(rational)}
\]

2. **Calculate \( 512 \cdot 4 \)**:
\[
512 \cdot 4 = 2048 \quad \text{(rational)}
\]

### Explanation of the Responses:
- **"Rational, because both numbers are rational."**: Correct—both \( 512 \) and \( \sqrt{16} \) are rational.
- **"Irrational, because both factors are irrational."**: Incorrect—\( \sqrt{16} \) is rational.
- **"Radical, because one factor is a radical."**: Misleading—describes the form, not the type of number resulting from the calculation.
- **"Irrational, because one factor is irrational."**: Incorrect—there are no irrational factors here.

### Conclusion:
The result \( 2048 \) is **rational**.

Answer 5

To determine the type of number from \( \frac{5}{12} \cdot \sqrt{16} \), let's analyze the components:

1. **Calculate \( \sqrt{16} \)**:
\[
\sqrt{16} = 4 \quad \text{(rational)}
\]

2. **Calculate \( \frac{5}{12} \cdot 4 \)**:
\[
\frac{5}{12} \cdot 4 = \frac{5 \cdot 4}{12} = \frac{20}{12} = \frac{5}{3} \quad \text{(rational)}
\]

### Explanation of the Responses:
- **"Rational, because both numbers are rational."**: This is correct. \( \frac{5}{12} \) is rational, and \( \sqrt{16} = 4 \) is also rational. The product of two rational numbers is rational.

- **"Irrational, because both factors are irrational."**: Incorrect. One factor, \( \sqrt{16} \), is rational.

- **"Radical, because one factor is a radical."**: This response describes the form but doesn't classify the type of number resulting from the calculation. The product is still rational, despite one factor being a radical.

- **"Irrational, because one factor is irrational."**: This is incorrect because \( \sqrt{16} \) is not an irrational number; it is a rational number.

### Conclusion:
The result \( \frac{5}{3} \) is **rational** because both \( \frac{5}{12} \) and \( \sqrt{16} \) are rational numbers.