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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
mathsExponentsExponents and Logarithm Arithmetics

Arithmetics with Exponents

Arithmetics with exponents, without evaluating the exponent, is explained. For example `a^m xx a^n = a^(m+n)`. The list of formulas are derived using the first principles of exponent.

These known results are given as a set of formulas. Students are advised to work them out quickly using the first principles. No need to memorize, and if the formulas are used repeatedly, over time, these can be recalled quickly.


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What is an exponent?

  • "base" repeatedly multiplied the "power" number of times
  • "base" repeatedly multiplied the "power" number of times
  • an open exhibition of components

The answer is, "base" repeatedly multiplied the "power" number of times.

What is the value of `2^5`?

  • by first principles of exponents `2^5 = 2 xx 2 xx 2 xx 2 xx 2`
  • by first principles of exponents `2^5 = 2 xx 2 xx 2 xx 2 xx 2`
  • `2xx5 = 10`

The answer is "`2^5 = 2 xx 2 xx 2 xx 2 xx 2`"

What is `3xx3xx3xx3`?

  • `3^4`
  • `3^4`
  • `3xx4`

The answer is "`3^4`"

We learned the first principles of exponents as, for any numbers `a` and `n`,
`a^n = a xx a xx a xx cdots (n text( times))`

In this topic, the above is used to understand some known results.

These known results are given as a set of formulas. Students are advised to work them out quickly using the first principles. No need to memorize, and if the formulas are used repeatedly, over time, these can be recalled quickly.

What is `2^4 xx 2^3`?

Note that, by first principles
`color(coral)(2^4 = 2 xx 2 xx 2 xx 2)`
`color(deepskyblue)(2^3 = 2 xx 2 xx 2)`

So, the given expression
`color(coral)(2^4)xx color(deepskyblue)(2^3)`
` = color(coral)(2 xx 2 xx 2 xx 2) xx color(deepskyblue)(2 xx 2 xx 2)`
The question is what is the simplified form of the expression given above?

  • the expression cannot be simplified
  • the simplified expression is `2^(4+3) = 2^7`
  • the simplified expression is `2^(4+3) = 2^7`

The answer is "the simplified expression is `2^(4+3) = 2^7`"

Generalizing this, for real numbers `a`, `m` and `n`,
`a^m xx a^n = a^(m+n)`

What is `2^4 xx 3^4`?

Note : by first principles
`color(coral)(2^4 = 2 xx 2 xx 2 xx 2)`
`color(deepskyblue)(3^4 = 3 xx 3 xx 3 xx 3)`

So, the given expression
`color(coral)(2^4)xx color(deepskyblue)(3^4)`
` = color(coral)(2 xx 2 xx 2 xx 2) xx color(deepskyblue)(3 xx 3 xx 3 xx 3)`
` = (color(coral)(2)xx color(deepskyblue)(3)) ``xx (color(coral)(2)xx color(deepskyblue)(3)) ``xx (color(coral)(2)xx color(deepskyblue)(3)) ``xx (color(coral)(2)xx color(deepskyblue)(3))`
The question is what is the simplified form of the expression given above?

  • the expression cannot be simplified
  • the simplified expression is `(2xx3)^4`
  • the simplified expression is `(2xx3)^4`

The answer is "`(2xx3)^4`"

Generalizing this, for real numbers `a`, `m` and `n`,
`a^m xx b^m = (a xx b)^m`

What is `2^4 -: 2^3`?

Note : by first principles
`color(coral)(2^4 = 2 xx 2 xx 2 xx 2)`
`color(deepskyblue)(2^3 = 2 xx 2 xx 2)`

So, the given expression
`color(coral)(2^4)-: color(deepskyblue)(2^3)`
` = color(coral)(2 xx 2 xx 2 xx 2) -: [color(deepskyblue)(2 xx 2 xx 2)]`
by properties of arithmetics
` = color(coral)(2 xx 2 xx 2 xx 2) xx [color(deepskyblue)(1/2 xx 1/2 xx 1/2)]`
The question is what is the simplified form of the expression given above?

  • the expression cannot be simplified
  • the simplified expression is `2^(4-3) = 2^1`
  • the simplified expression is `2^(4-3) = 2^1`

The answer is "`2^(4-3) = 2^1`"

Generalizing this, for real numbers `a`, `m` and `n`,
`a^m -: a^n = a^(m-n)`

What is `2^4 -: 3^4`?

Note : by first principles
`color(coral)(2^4 = 2 xx 2 xx 2 xx 2)`
`color(deepskyblue)(3^4 = 3 xx 3 xx 3 xx 3)`

So, the given expression
`color(coral)(2^4)-: color(deepskyblue)(3^4)`
` = color(coral)(2 xx 2 xx 2 xx 2) -: color(deepskyblue)(3 xx 3 xx 3 xx 3)`
` = color(coral)(2 xx 2 xx 2 xx 2) xx color(deepskyblue)(1/3 xx 1/3 xx 1/3 xx 1/3)`
` = color(coral)(2)/color(deepskyblue)(3) xx color(coral)(2)/color(deepskyblue)(3) ``xx color(coral)(2)/color(deepskyblue)(3) xx color(coral)(2)/color(deepskyblue)(3)`
The question is, what is the simplified form of the expression given above?

  • the expression cannot be simplified
  • the simplified expression is `(2/3)^4`
  • the simplified expression is `(2/3)^4`

The answer is "`(2/3)^4`"

Generalizing this, for real numbers `a`, `m` and `n`,
`a^m -: b^m = (a -: b)^m`

What is the value of `3^0`?

Note that
`3^0`
` = 3^(1-1)`
` = 3^1 -: 3^1`
` = 3/3`
The question is, what is the value of the expression.

  • `1`
  • `1`
  • `0`

The answer is "1".

Generalizing this, for a real number `a`,
`a^0 = 1`

What is `(5^2)^3`?

Note that by first principles,
`(5^2)^3`
`=(5^2)xx(5^2)xx(5^2)`
substituting `5^2 = 5 xx 5`
`= (5xx5)xx(5xx5)xx(5xx5)`

The question is, what is the simplified form of the expression given above?

  • the expression cannot be simplified
  • the simplified expression is `5^(2 xx3) = 5^6`
  • the simplified expression is `5^(2 xx3) = 5^6`

The answer is "`5^(2 xx3) = 5^6`."

Generalizing this, for real numbers `a`, `m` and `n`,
`(a^m)^n = a^(mn)`

What is `2^3 + 2^3 + 2^3 + 2^3 + 2^3`?

Note, by first principles a number repeatedly added is a multiplication.

  • `5 xx 2^3`
  • `5 xx 2^3`
  • `2^15`

The answer is "`5 xx 2^3`"

Generalizing this, for real numbers `a`, `m` and `n`,
`a^m + a^m + cdots (n text(times)) = nxxa^m`

What is `3 xx 2^5 + 4 xx 2^5`?

Note that by first principles,
`color(coral)(3 xx 2^5 = 2^5 + 2^5 + 2^5)`
`color(deepskyblue)(4 xx 2^5 = 2^5 + 2^5 + 2^5 + 2^5)`

So,
`color(coral)(3 xx 2^5) ``+ color(deepskyblue)(4 xx 2^5)`
`=color(coral)(2^5 + 2^5 + 2^5) ``+ color(deepskyblue)(2^5 + 2^5 + 2^5 + 2^5)`

The question is, what is the simplified form of the above expression?

  • `(3+4) xx 2^5 = 7 xx 2^5`
  • `(3+4) xx 2^5 = 7 xx 2^5`
  • addition of exponents cannot be simplified

The answer is "`(3+4) xx 2^5 = 7 xx 2^5`".

Generalizing this, for real numbers `a`, `m`, `n`, `p` and `q`,
`p xx a^m + q xx a^m = (p+q) xx a^m`

Known results in Exponents :
For `a, b, p, q, m, n in RR`,

`a^m xx a^n = a^(m+n) `

`a^m xx b^m = (a xx b)^m`

`a^m -: a^n = a^(m-n) `

`a^m -: b^m = (a -: b)^m`

`a^0 = 1`

`(a^m)^n = a^(mn)`

`a^(1/m) = root(m)(a)`

`a^(−m) = 1/(a^m)`

`a^m + a^m + cdots (n times) = nxxa^m`

`p xx a^m + q xx a^m = (p+q) xx a^m`



Solved Exercise Problem:

What is `3^(-2)`?

  • `9`
  • `1/9`
  • `1/9`

The answer is "`1/9`"

                            
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