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

Question

?(1 point)
Responses

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

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

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

Radical, because one factor is a radical.

Answer 1

To determine the type of number that results from \( 512 \cdot \sqrt{16} \), we can evaluate the expression step by step.

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

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

3. Multiply the two numbers: \[ 512 \cdot 4 = 2048 \]

Now, \( 2048 \) is a whole number and is also classified as a rational number (since rational numbers include all integers, whole numbers, and fractions).

Given this analysis, the correct answer is:

Rational, because both numbers are rational.

Answer 2

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

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

2.40; a rational number
2.40; a rational number

0.3726 . . . ; an irrational number
0.3726 . . . ; an irrational number

0.37; a rational number

Answer 3

To find the product of \( 5 - \sqrt{5} \) and \( 16 \), we first need to calculate the value of \( 5 - \sqrt{5} \):

1. Calculate \( \sqrt{5} \): The exact value of \( \sqrt{5} \) is approximately \( 2.236 \).

2. Subtract \( \sqrt{5} \) from 5: \[ 5 - \sqrt{5} \approx 5 - 2.236 = 2.764 \]

3. Multiply by 16: \[ (5 - \sqrt{5}) \cdot 16 \approx 2.764 \cdot 16 = 44.224 \]

Now that we have found the product, we need to determine its type:

• \( 5 - \sqrt{5} \) is not an integer, and since \( \sqrt{5} \) is irrational, \( 5 - \sqrt{5} \) is also irrational.
• Therefore, the product \( 16(5 - \sqrt{5}) \) is also irrational, as the product of a rational number (16) and an irrational number \( (5 - \sqrt{5}) \) is irrational.

Given the options available, the most appropriate choice is:

2.4027 . . . ; an irrational number.

However, note that the numerical value of \( 44.224 \) does not match \( 2.4027 \)... So the numerical answer should be adjusted to reflect the actual calculations, but in terms of classification, it is certainly an irrational number.

Answer 4

What index should the root form of the expression 1258⋅12−12
have?(1 point)

Answer 5

To determine the appropriate index for the root form of the expression \( 1258 \cdot 12^{-12} \), we need to analyze each component of the expression.

1. The number \( 1258 \) is a positive integer.
2. The term \( 12^{-12} \) represents the reciprocal of \( 12^{12} \), which can also be written as \( \frac{1}{12^{12}} \).

The expression can be rewritten in a root form as follows:

\[ 1258 \cdot 12^{-12} = 1258 \cdot \frac{1}{12^{12}} = \frac{1258}{12^{12}} \]

To express this in root form, we can use the properties of exponents. The expression \( \frac{1}{12^{12}} \) can be rewritten as a root:

\[ \frac{1}{12^{12}} = \frac{1}{(12^{12})} = 12^{-12} = \frac{1}{\sqrt[12]{12^{12}}} \]

Thus, the index of the root would be 12 due to the exponent of 12 in the denominator.

In conclusion, the index for the root form of the expression \( 1258 \cdot 12^{-12} \) should be:

12.

Answer 6

What index should the root form of the expression 12^5/8⋅12−1/2
have?(1 point)

Answer 7

To determine the appropriate index for the root form of the expression \( \frac{12^5}{8} \cdot 12^{-\frac{1}{2}} \), let's break it down step-by-step:

1. Rewriting the Expression: We can combine the terms in the expression: \[ \frac{12^5}{8} \cdot 12^{-\frac{1}{2}} = \frac{12^5 \cdot 12^{-\frac{1}{2}}}{8} \]

2. Combining the Exponents: By using the property of exponents \( a^m \cdot a^n = a^{m+n} \), we can combine the exponents: \[ 12^{5 - \frac{1}{2}} = 12^{5 - 0.5} = 12^{4.5} = 12^{\frac{9}{2}} \]

3. Revisiting the Expression: Now, we rewrite the entire expression with \( 12^{\frac{9}{2}} \): \[ \frac{12^{\frac{9}{2}}}{8} \]

4. Expressing by Roots: The term \( 12^{\frac{9}{2}} \) can be rewritten in root form: \[ 12^{\frac{9}{2}} = \sqrt{12^9} = \sqrt[2]{12^9} \]

From the expression \( \sqrt{12^{9}} \), we can identify the root index:

• The root indicates the index of the square root, which is 2 since it represents the square root.

Alternatively, if we consider the full expression \( \frac{12^{\frac{9}{2}}}{8} \) directly and note \( 8 \) doesn’t change the index, the more simplified version in terms of \( 12^{\frac{9}{2}} \) retains the index of 2.

Thus, the index for the root form of the expression \( \frac{12^5}{8} \cdot 12^{-\frac{1}{2}} \) is:

2.