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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

This page introduces the the properties of exponents. To understand polynomials and equations in algebra, these properties are used.



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Which of the following equals `p xx p xx p ... (m text( times))`?

  • `p^m`
  • `p^m`
  • `m p`

The answer is `p^m`

Which of the following equals `p^m xx q^m`?

  • `mab`
  • `(p xx q)^m`
  • `(p xx q)^m`

The answer is `(p xx q)^m`.

`p^m xx q^m`

`=p xx p xx ... (m text( times)) ``xx q xx q xx ... (m text( times))`

`=pq xx pq xx ... (m text( times)) `

`=(pq)^m`

Which of the following equals `p^m -: p^n`?

  • `p^(m-n)`
  • `p^(m-n)`
  • `p^(m/n)`

The answer is `p^(m-n)`

Which of the following equals `p^m -: q^m`?

  • `p/q`
  • `(p -: q)^m`
  • `(p -: q)^m`

The answer is `(p -: q)^m`

Which of the following equals `p^0`?

  • `0`
  • `1`
  • `1`

The answer is `1`.

`p^0= p^(1-1) = p/p =1`

Which of the following equals `(p^m)^n`?

  • `p^(m^n)`
  • `p^(mn)`
  • `p^(mn)`

The answer is `p^(mn)`

Which of the following equals `root(m)(p)`?

  • `p^(1/m)`
  • `p^(1/m)`
  • `p^(-m)`

The answer is `p^(1/m)`

Which of the following equals `p^(-m)`?

  • `1/(p^m)`
  • `1/(p^m)`
  • `-p^m`

The answer is `1/(p^m)`

Which of the following equals `p^m + p^m + …(n text ( times))`?

  • `np^m`
  • `np^m`
  • `p^(mn)`

The answer is `np^m`

Which of the following equals `k xx p^m + l xx p^m`?

  • `(k+l)xx p^m`
  • `(k+l)xx p^m`
  • `kl p^m`

The answer is `(k+l)xx p^m`

Which of the following equals `k xx p^m + l xx p^n`?

  • `(k+l)xx p^m`
  • it cannot be simplified
  • it cannot be simplified

The answer is : it cannot be simplified. Expressions of this type lead to the definition of algebraic expressions and polynomials in the form `ax^m+bx^n+c`.

Which of the following equals `(p+q)^m`?

  • `p^m+q^m`
  • it cannot be simplified
  • it cannot be simplified

The answer is : it cannot be simplified. Expressions of this type lead to the definition of algebraic identities.

Properties of Exponents
     `a^m + a^m + ... (n text( times)) = n a^m`
     `pa^m + qa^m = (p+q)a^m`
     `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^m)^n = a^(mn)`
     `a^(1/m) = root(m)(a)`
     `a^(-m) = 1/(a^m)`
     `log_a a^m = m`

 •  Some forms cannot be simplified any further
That is, the expressions can be evaluated to equivalent numerical values, but not simplified retaining the power of `a`

    →  `pa^m + qa^n + r` cannot be simplified for `m != n`
This is the basis for algebraic expressions and polynomials

    →  `(a+b)^m` cannot be simplified, but an equivalent expression can be defined in the general form.
This is the basis for algebraic identities.

                            
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