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Thought-Process to Discover Knowledge

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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.

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mathsExponentsFundamentals of Exponents

### root : An inverse of Exponent

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.

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What is the inverse of addition?

• subtraction
• subtraction
• division

If the sum and the first addend are given, then

If the sum and the second addend are given, then

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

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

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

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