Exponents of Negative Numbers

Squaring Removes Any Negative

"Squaring" means to multiply a number by itself.

  • Squaring a positive number gets a positive effect: (+5) × (+5) = +25
  • Squaring a negative number also gets a positive result: (−five) × (−5) = +25

Considering a negative times a negative gives a positive. So:

5x5 = -5x-5

"So what?" you say ...

... well take a await at this:

Square root of square

Oh no! Nosotros started with minus 3 and concluded with plus 3.

When we square a number, then take the square root, we may not stop up with the number nosotros started with!

In fact we stop up with the absolute value of the number:

√(x two ) = |ten|

That also happens for all fifty-fifty (but not odd) Exponents.

Try here:

images/exponent-calc.js

Even Exponents of Negative Numbers

An even exponent always gives a positive (or 0) result.

That simple fact tin can brand our life easier:

1 (Odd): (−ane)1 = −1

2 (Even): (−1)2 = (−1) × (−1) = +i

3 (Odd): (−1)iii = (−1) × (−i) × (−1) = −i

4 (Even): (−i)4 = (−i) × (−1) × (−one) × (−1) = +1

Practice y'all see the −i, +i, −1, +1 blueprint?

(−one)odd = −1

(−1)even = +1

And so nosotros tin "shortcut" some calculations, like:

Instance: What is (−1)97 ?

97 is odd, and so:

(−1)97 = −1

Example: What is (−two)6 ?

two6 = 64, and six is even, and then:

(−2)half-dozen = +64

Roots of Negative Numbers

Example: What is the value of 10 here: x2 = −1

Does x=1?

one × 1 = +1

Does x=−i?

(−i) × (−1) = +ane

We can't get −one for an respond!

Information technology seems incommunicable!

Well, it is impossible using Real Numbers.

Only we can practice it using Imaginary Numbers.

In other words:

√−1 is not a Existent Number ...

... it is an Imaginary Number

This is truthful for all fifty-fifty roots:

An Even Root of a Negative Number is Not Real

And so merely exist conscientious when taking foursquare roots, quaternary roots, 6th roots, etc.

1742, 3998, 459, 3999, 460, 1743, 1093, 4000, 1094, 4001