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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.
mathsFoundation of Algebra with Numerical ArithmeticsNumerical Arithmetics: Laws and Properties for Algebra

### Laws and Properties of Arithmetics : Exponents

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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