nubtrek

Server Error

Server Not Reachable.

This may be due to your internet connection or the nubtrek server is offline.

Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue
mathsExponentsFundamentals of Exponents

root : An inverse of Exponent

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