This page introduces "root".

One of the inverses of exponent is root. Root is introduced with the following two.

• first principle -- Root of a number to a given power of root is the base of the exponent with the given power.

• Simplified Procedure -- Root of a number is found from prime-factorization of the numbers (if root evaluates to an integer). This introduction "root is an inverse of exponent" is *astoundingly clear and makes it simple for students*.

*click on the content to continue..*

What is the inverse of addition?

- subtraction
- subtraction
- division

The answer is "subtraction".

Addition is

first addend + second addend = sum.

If the sum and the first addend are given, then

second addend = sum - first addend

If the sum and the second addend are given, then

first addend = sum - second addend

Subtraction serves as the inverse of addition for both addends.* subtraction is the inverse for both because addition is commutative `3+2 = 5` and `2+3=5`. *

What is the inverse of multiplication?

- subtraction
- division
- division

The answer is "division"

Multiplication is

multiplicand `xx` multiplier `=` product.

If the multiplicand and the product are given, then

multiplier `=` product `-:` multiplicand

If the multiplier and the product are given, then

multiplicand `=` product `-:` multiplier

Division serves as the inverse of multiplication for both multiplier and multiplicand.* Division is the inverse for both because, multiplication is commutative. `3xx2 = 5` and `2xx3=5`. *

What is the inverse of "exponent"?

- given result of exponentiation and base, find the power
- given result of exponentiation and power, find the base
- both the above
- both the above

The answer is "both the above"

Two inverses are defined for exponents.*The exponent is not commutative. `3^2=9` and `2^3=8` `a^b !=b^a` *

Exponent is

`(text(base))^(text(power)) = text(exp. result)`

if exponentiation result and power are given, then

`text(base) = root(text(power))(text(result))`

This is called "root".

The same in another form is

`text(base) = (text(exp. result))^(1/text(power))`

This is exponent to a fraction.

If exponentiation-result and base are given, then

`text(power) = log_text(base) (text(exp.result))`

This inverse is called "logarithm".

Which of the following is a meaning for the word "root"?

- basic source or origin of something
- basic source or origin of something
- short form of kangaroo

The answer is "basic source or origin of something".

What is the term used to refer finding base from the exponentiation result?

- Pronunciation : Say the answer once

Spelling: Write the answer once

The answer is "root".

**Roots** : Root of a number to a given power of root is the base of the exponent with the given power.

`root(3)(8) = 2`

`3` is the power of root

`8` is the number for which root is calculated

`2` is the result of root

`root(3)(8) = 2` implies that `2^3 = 8`, and *the operation root finds the base in the equivalent exponent*.

**Finding Root (First Principles)** : Root of a number is the base in the equivalent exponent.

eg: `root(3)(64)` is seen as the exponent `64=4^3`. The base is `4` and so `root(3)(64) = 4`

Which of the following is `81^(1/4)`

- `root(4)(81)`
- `root(4)(81)`
- `log_4 81`

The answer is "`root(4)(81)`"

Find `root(3)(125)`

- `41 2/3`
- `5`
- `5`

The answer is "`5`".

To find `root(3)(125)`, perform prime-factorization on the given value.

`125=5xx5xx5`

From this, it is evident that `125=5^3`. By first principles, `root(3)(125)=5`

Find `root(2)(36)`

- `23`
- `6`
- `6`

The answer is "`6`".

To find `root(2)(36)`, perform prime-factorization on the given value.

`36=2xx2xx3xx3`

re-arrage such that the factors are grouped

`36=(2xx3)xx(2xx3)`

There are two groups equal to the power of the root `2`.

pick one group and compute the result.

By first principles, `root(2)(36)= 2 xx 3 = 6`

**Finding Roots (Simplified Procedure)** : To find roots of a number, express the number in prime factors and group the factors.

eg: `root(3)(1000)` `=root(3)(2xx2xx2xx5xx5xx5)` `=10`

*slide-show version coming soon*