Math

QuestionIdentify the number set(s) for 16\sqrt{-16}. Choose all that apply: real, complex, rational, imaginary, irrational.

Studdy Solution

STEP 1

Assumptions1. We are asked to identify the set(s) of numbers to which the number 16\sqrt{-16} belongs. . The sets of numbers we are considering are real numbers, complex numbers, rational numbers, imaginary numbers, and irrational numbers.

STEP 2

To determine the set(s) to which 16\sqrt{-16} belongs, we first need to simplify 16\sqrt{-16}.

STEP 3

The square root of a negative number is not a real number, but an imaginary number. We can express the square root of a negative number as the square root of the absolute value of the number times ii, where ii is the imaginary unit with the property that i2=1i^2 = -1.
16=16×i\sqrt{-16} = \sqrt{16} \times i

STEP 4

implify 16\sqrt{16}.
16=4\sqrt{16} =4So, 16=4i\sqrt{-16} =4i.

STEP 5

Now that we have simplified 16\sqrt{-16} to 4i4i, we can identify the set(s) to which 4i4i belongs.

STEP 6

4i4i is not a real number, because real numbers are all numbers that can be located on the number line, including both rational and irrational numbers. Since 4i4i cannot be located on the number line, it is not a real number.

STEP 7

4i4i is a complex number, because complex numbers include all numbers that can be expressed in the form a+bia + bi, where aa and bb are real numbers and ii is the imaginary unit.

STEP 8

4i4i is not a rational number, because rational numbers are all numbers that can be expressed as a fraction a/ba/b, where aa and bb are integers and b0b \neq0. Since 4i4i cannot be expressed in this form, it is not a rational number.

STEP 9

4i4i is an imaginary number, because imaginary numbers are all numbers that can be expressed in the form bibi, where bb is a real number and ii is the imaginary unit.

STEP 10

4i4i is not an irrational number, because irrational numbers are all numbers that cannot be expressed as a fraction a/ba/b, where aa and bb are integers and b0b \neq0, and that are not imaginary numbers. Since 4i4i is an imaginary number, it is not an irrational number.
So, 16\sqrt{-16} belongs to the set of complex numbers and the set of imaginary numbers.

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