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

Question

?
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

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 2

a little shorter

Answer 3

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 4

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 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.