how to find the square root of a negative number

Let's investigate what happens when negative values appear nether the radical symbol (as the radicand) for cube roots and foursquare roots.

In some situations, negative numbers nether a radical symbol are OK. For instance, is not a problem since (-2) • (-ii) • (-ii) = -eight, making the answer -2. In cube root problems, it is possible to multiply a negative value times itself three times and get a negative answer.

Difficulties, however, develop when we look at a trouble such as . This square root problem is request for a number multiplied times itself that will requite a production (answer) of -16. At that place just is no way to multiply a number times itself and get a negative event. Consider: (iv) • (4) = 16 and (-4) • (-four) = sixteen.

CUBE ROOTS:	Only	Foursquare ROOTS:
Yes, (-2) x (-two) 10 (-2) = -8. No problem.		Nope! (4) ten (4) ≠ -16. Nope! (-4) 10 (-iv) ≠ -16.

Foursquare roots are the culprits! The difficulties arise when you see a negative value nether a square root. Information technology is not possible to square a value (multiply it times itself) and go far at a negative value. So, what do we do?

The foursquare root of a negative number does non be among the set of Real Numbers.

When bug with negatives under a square root starting time appeared, mathematicians thought that a solution did non exist. They saw equations such as ten ² + ane = 0 , and wondered what the solution really meant.

In an effort to address this problem, mathematicians "created" a new number,

i , which was referred to as an "imaginary number" , since it was not in the set of "Real Numbers". This new number was viewed with much skepticism. The imaginary number kickoff appeared in print in the year 1545.

The imaginary number "i" is the foursquare root of negative i.

An imaginary number possesses the unique property that when squared, the result is negative.

irad1

Consider:
The process of simplifying a radical containing a negative gene is the same as normal radical simplification. The only difference is that the volition be replaced with an " i ".

Equally enquiry with imaginary numbers continued, information technology was discovered that they actually filled a gap in mathematics and served a useful purpose. Imaginary numbers are essential to the study of sciences such equally electricity, breakthrough mechanics, vibration analysis, and cartography.

When the imaginary i was combined with the set of Real Numbers, the all encompassing set of Circuitous Numbers was formed.

complex

Product Dominion

where a ≥ 0, b≥ 0

"The foursquare root of a product is equal to the product of the foursquare roots of each factor."

This theorem allows usa 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

FYI: The powers of i e'er cycle through merely iv dissimilar values:
i ^one = i
i ² = -1
i ³ = -i
i ⁴ = 1
i ⁵ = i and the cycle starts once more.

When doing arithmetic on i , treat it as you would an "x".
3 i + four i = seven i
2 i • 4 i = 8 i ⁱⁱ = -eight
Annotation: i ² was replaced with -ane.

Do Not confuse "irrational" numbers
with
"imaginary" numbers.
They are Not the aforementioned.

Complex Numbers
a ± bi
where a and b are real numbers, and i is the imaginary unit.

Complex numbers are written in the standard course a + bi.

In Algebra ane, you volition see that the "imaginary" number will be useful when solving quadratic equations. The quadratic formula may requite circuitous solutions, written as a ± bi, where a and b are real numbers.
You volition see more about imaginary numbers in the Quadratic section.

Source: https://mathbitsnotebook.com/Algebra1/Radicals/RADNegativeUnder.html

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