6 4 Practice Nth Roots

nth Root

The "nth Root" used n times in a multiplication gives the original value

" nth ? "

1st, 2nd, iiird, ivth, 5th, ... due northth ...

Instead of talking well-nigh the "fourth", "16th", etc, we can merely say the " nth ".

The nth Root

The "2nd" root is the foursquare root
The "third" root is the cube root
etc!

2	√ a × √ a = a	The square root used two times in a multiplication gives the original value.
3	3 √ a × 3 √ a × iii √ a = a	The cube root used three times in a multiplication gives the original value.

due north	n √ a × northward √ a × ... × n √ a = a (northward of them)	The nth root used n times in a multiplication gives the original value.

And then it is the general mode of talking near roots
(so it could be 2nd, or 9th, or 324th, or whatever)

The nth Root Symbol

This is the special symbol that means "nth root", information technology is the "radical" symbol (used for foursquare roots) with a little n to mean nth root.

Using information technology

Nosotros could use the nth root in a question like this:

Question: What is "northward" in this equation?

n √ 625 = five

Answer: I just happen to know that 625 = 5⁴ , and then the 4thursday root of 625 must exist 5:

4 √ 625 = 5

Or we could use "n" considering we want to say general things:

Example: When n is odd then n √ aⁿ = a (nosotros talk nearly this afterward).

Why "Root" ... ?

When you see "root" think

"I know the tree , just what is the root that produced information technology? "

Case: in √nine = 3 the "tree" is ix , and the root is iii .

Backdrop

Now nosotros know what an nth root is, let us look at some properties:

Multiplication and Division

We tin can "pull apart" multiplications under the root sign like this:

n √ ab = n √ a × north √ b
(Note: if n is even so a and b must both be ≥ 0)

This can help us simplify equations in algebra, and likewise make some calculations easier:

Example:

iii √ 128 = 3 √ 64×2 = 3 √ 64 × iii √ 2 = 43 √ two

So the cube root of 128 simplifies to four times the cube root of 2.

It besides works for division:

northward √ a/b = n √ a / n √ b
(a≥0 and b>0)
Notation that b cannot be zero, equally nosotros tin't split by goose egg

Case:

3 √ 1/64 = 3 √ 1 / 3 √ 64 = one/4

And then the cube root of 1/64 simplifies to but i quarter.

Addition and Subtraction

Just we cannot do that kind of thing for additions or subtractions!

northward √ a + b ≠ due north √ a + n √ b

northward √ a − b ≠ due north √ a − n √ b

due north √ aⁿ + b^northward ≠ a + b

Example: Pythagoras' Theorem says

a² + b² = c^two

And so we summate c like this:

c = √ a² + b^two

Which is non the same every bit c = a + b , right?

It is an like shooting fish in a barrel trap to autumn into, so beware.

It also ways that, unfortunately, additions and subtractions tin be difficult to deal with when under a root sign.

Exponents vs Roots

An exponent on ane side of "=" can be turned into a root on the other side of "=":

If aⁿ = b so a = n √ b

Notation: when n is even then b must be ≥ 0

Case:

5⁴ = 625 so 5 = 4 √ 625

nth Root of a-to-the-nth-Power

When a value has an exponent of due north and nosotros take the nth root we get the value back once more ...

... when a is positive (or zero):

(when a ≥ 0 )

Example:

... or when the exponent is odd :

(when north is odd )

Instance:

... but when a is negative and the exponent is fifty-fifty we become this:

Did you encounter that −3 became +iii ?

... so nosotros must do this:

(when a < 0 and due north is even )

The |a| means the absolute value of a, in other words any negative becomes a positive.

Example:

And so that is something to be careful of! Read more than at Exponents of Negative Numbers

Here it is in a lilliputian table:

nth Root of a-to-the-mth-Power

What happens when the exponent and root are different values (m and n)?

Well, we are allowed to change the order similar this:

n √ a^m = ( n √ a )^m

So this: nth root of (a to the power grand)
becomes (nth root of a) to the power m

Example:

3 √ 27^two = ( three √ 27 )²
= 3²
= 9

Easier than squaring 27 so taking a cube root, right?

But there is an even more powerful method ... we can combine the exponent and root to brand a new exponent, like this:

n √ a^m = a ^{grand north}

The new exponent is the fraction 1000 north which may be easier to solve.

Example:

3 √ 4⁶ = 4 ^{half dozen 3}
= 4²
= sixteen

This works considering the nth root is the aforementioned equally an exponent of (1/n)

north √ a = a ^{ane northward}

You might like to read nigh Fractional Exponents to detect out why!

318, 2055, 319, 317, 1087, 2056, 1088, 2057, 3159, 3160