*

Negative Value Underthe Square Root Radical ubraintv-jp.com Topical Overview | Algebra 1 Summary | MathBits" Teacher Reresources Terms of Use Contact Person: Donna Roberts
*

Let"s investigate what happens as soon as negative values appear under the radical symbol (as the radicand) for cube roots and square roots.

You are watching: What is the square root of negative 2

In some cases, negative numbers under a radical symbol are OK. For instance,

*
is not a trouble given that (-2) • (-2) • (-2) = -8, making the answer -2. In cube root troubles, it is feasible to multiply an unfavorable value times itself three times and also gain a negative answer.

Difficulties, however, develop when we look at a problem such as

*
. This square root difficulty is asking for a number multiplied times itself that will certainly offer a product (answer) of -16. There simply is no method to multiply a number times itself and gain an unfavorable outcome. Consider: (4) • (4) = 16 and also (-4) • (-4) = 16.


CUBE ROOTS:
*

BUT
SQUARE ROOTS:
*

Yes, (-2) x (-2) x (-2) = -8. No problem.

Nope! (4) x (4) ≠ -16. Nope! (-4) x (-4) ≠ -16.


*
Square roots are the culprits! The challenges arise when you encounter an adverse worth under a square root. It is not possible to square a worth (multiply it times itself) and also arrive at a negative value. So, what carry out we do?


*

The square root of a negative number does not exist among the set of Real Numbers.

When difficulties via negatives under a square root initially appeared, mathematicians assumed that a solution did not exist. They observed equations such as x2 + 1 = 0, and also wondered what the solution

*
really supposed. In an initiative to deal with this trouble, mathematicians "created" a brand-new number, i, which was described as an "imaginary number", since it was not in the collection of "Real Numbers". This brand-new number was viewed with much skepticism. The imaginary number initially appeared in print in the year 1545.

*
The imaginary number "i" is the square root of negative one.

An imaginary number possesses the distinct property that when squared, the outcome is negative.

*

Consider:

*
The process of simplifying a radical containing an adverse variable is the very same as normal radical simplification. The only distinction is that the
*
will certainly be reinserted via an "i ".

As research with imaginary numbers ongoing, it was found that they actually filled a gap in mathematics and also offered a valuable purpose. Imaginary numbers are vital to the research of scientific researches such as electricity, quantum mechanics, vibration analysis, and also cartography.

When the imaginary i was combined via the collection of Real Numbers, the all encompassing set of Complex Numbers was formed.

See more: How Do You Spell Tear As In Rip ? Tear Vs Tear

*


Product Rule wbelow a ≥ 0, b≥ 0

"The square root of a product is equal to the product of the square roots of each aspect."

This theorem permits us to use our method of simplifying radicals.


Imaginary (Unit) Number
*

Product Rule (extended) where a ≥ 0, b≥ 0 OR a ≥ 0, b 0 but NOT a 0, b 0
*