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Thought-Process to Discover Knowledge

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mathsExponentsNumerical Expressions with Exponents

Exponents: Simplification of Expressions

In this page, handing of exponents, roots, and logarithms in numerical expressions is explained.

The precedence order PEMDAS / BODMAS is explained.

For operations of same precedence order, the sequence of operation "simplification from left to right" is explained.


click on the content to continue..

Let us quickly revise what is numerical expression, and precedence order, sequence in simplifying the numerical expressions. This was introduced in whole numbers, reviewed in integers, reviewed again in fractions, reviewed again in decimals.

The precedence order was given as BODMAS or PEMDAS. The "O" in BODMAS or "E" in PEMDAS were not explained.

Let us quickly revise the fundamentals required, and get to the explanation of "O" in BODMAS or "E" in PEMDAS.

`O` stands for order, representing exponents, root, and logarithm.
`E` stands for exponents, representing all of exponents, root, and logarithm.

What is `2+4+3`?

  • `9`
  • `9`
  • `243`

The answer is "`9`".

This is an example of a numerical expression.

Which of the following is a meaning for the word "expression"?

  • something done very fast
  • collection of numbers and arithmetic operations between them, which together represent a quantity
  • collection of numbers and arithmetic operations between them, which together represent a quantity

The answer is "collection of numbers and arithmetic operations between them, which together represent a quantity".

What is `2xx4xx3`?

  • `24`
  • `24`
  • `243`

The answer is "`24`".

This is an example of a numerical expression.

Is `a+3x` a numerical expression?

  • Yes. It has the number 3.
  • No. It is not entirely numbers and arithmetic operations
  • No. It is not entirely numbers and arithmetic operations

The answer is "No. It is not entirely numbers and arithmetic operations".

Is `3+4xx2-6-:3` a numerical expression?

  • No, as it has the addition, subtraction, multiplication, division
  • Yes, all arithmetic operations can be part of a numerical expression
  • Yes, all arithmetic operations can be part of a numerical expression

The answer is "Yes, all arithmetic operations can be part of a numerical expression"

Consider `1+2` and `3xx1`. Note that when evaluated, both result in identical numerical value `1+2=3` and `3xx1 = 3`.

Are these two different numerical expressions?

  • They are two different expressions, evaluating to equal values
  • They are two different expressions, evaluating to equal values
  • They are identical expressions as they evaluate to equal values

The answer is "They are two different expressions, evaluating to equal values".

What is the rule of precedence in numerical expressions?

  • multiplication and division have higher precedence over addition and subtraction
  • parentheses or brackets have the highest precedence
  • both the above
  • both the above

The answer is "both the above".

Simplify `9-6-:3`

  • `7`
  • `7`
  • `1`

The answer is "`7`". Division has higher precedence over subtraction. So

`9-6-:3`

`=9-2`

`=7`

What is the rule of sequence in numerical expressions?

  • there is no such thing as rule of sequence
  • when multiple operation of same precedence is to be simplified, the operations are performed from left to right sequence
  • when multiple operation of same precedence is to be simplified, the operations are performed from left to right sequence

The answer is "when multiple operation of same precedence is to be simplified, the operations are performed from left to right sequence".

Simplify `6-:3xx2`

  • `4`
  • `4`
  • `1`

The answer is "`4`".

The division and multiplication are of same precedence, so it is simplified from left to right.

`6-:3xx2`

`=2xx2`

`=4`

What is the rule of brackets or parentheses in numerical expressions?

  • there is no such thing as rule of brackets
  • the subexpression within a bracket or parentheses has the highest precedence
  • the subexpression within a bracket or parentheses has the highest precedence

The answer is "the subexpression within a bracket or parentheses has the highest precedence".

Simplify `6-:(3xx2)`

  • `4`
  • `1`
  • `1`

The answer is "`1`".

The bracket has higher precedence, and so the expression inside bracket is simplified first.

`6-:(3xx2)`

`=6-:6`

`=1`

In a numerical expression, the precedence order is:

 •  Parentheses or Brackets are at the highest precedence order

 •  Exponents, roots, logarithms are of next highest precedence order. These three are in the same level of precedence.

 •  Division and multiplication are the next, and these two are of same level of precedence.

 •  Addition and subtraction are the last, and these two are of same level of precedence.

This is abbreviated as BODMAS (Brackets, Order, Division, Multiplication, Addition, Subtraction) or PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction).

When multiple operation of same precedence is to be simplified, the operations are performed from left to right sequence.

Simplify `11-3^2`

  • `64`
  • `2`
  • `2`

The answer is "`2`".

The exponent is of higher precedence than subtraction.

`11-3^2`

`=11-9`

`=2`

Simplify `log_3 9 + 18`

  • `20`
  • `20`
  • `3`

The answer is "`20`".

The logarithm is of higher precedence than addition.

`log_3 9 + 18`

`=2+18`

`=20`

Simplify `4xx4^(1/2)`

  • `8`
  • `8`
  • `4`

The answer is "`8`".

The square root is of higher precedence than subtraction.

`4xx4^(1/2)`

`=4 xx 2`

`=8`

Simplify `log_2 4^3`

Which one of the following is correct?

`color(coral)(log_2 4^3)`
`color(coral)(=2^3)`
`color(coral)(=8)`

`color(deepskyblue)(log_2 4^3)`
`color(deepskyblue)(=log_2 (2^6))`
`color(deepskyblue)(=6)`

  • `8`
  • `6`
  • `6`

The answer is "`6`".

The notation of cube of log is `log_2^3 4`. The notation for log of a cube is `log_2 4^3`.

`color(coral)(log_2^3 4)`
logarithm is applied first
`color(coral)(=2^3)`
`color(coral)(=8)`

`color(deepskyblue)(log_2 4^3)`
exponent is applied first
`color(coral)(=log_2 (2^6))`
`color(coral)(=6)`

Simplify `4-6-:3^(-2)`

  • `3.75`
  • `-50`
  • `-50`

The answer is "`-50`".

`4-6-:3^(-2)`

exponent is of higher precedence
`=4-6 -:(1/9)`

division is of higher precedence
`=4-54`

`=-50`.

Simplify `log_10 100 + 9900`.

  • `9902`
  • `9902`
  • `3`

The answer is "`9902`".

`log_10 100 + 9900`

The logarithm is of higher precedence over addition
`=2+9900`

`=9902`

Solved Exercise Problem:

Simplify `root(3)((-1+3-1))+(-2)-1`

  • `-5`
  • `-2`
  • `-2`

The answer is "`-2`"

`root(3)((-1+3-1))+(-2)-1`

Bracket is of the highest precedence
`=root(3)(1)+(-2)-1`

exponent or root is higher in precedence
`=1+(-2)-1`

left to right sequence for operations of same precedence
`=-1-1`

`=-2`

Simplification of Expressions : BODMAS

 •  B - Brackets

 •  O - Order (exponents, roots, logarithm)

 •  D - Division

 •  M - Multiplication

 •  A - Addition

 •  S - Subtraction

 •  And Left to Right sequence for multiple operations of same precedence.

PEMDAS

 •  P - Parentheses

 •  E - Exponents (roots and logarithm)

 •  M - Multiplication

 •  D - Division

 •  A - Addition

 •  S - Subtraction

 •  And Left to Right sequence for multiple operations of same precedence.

                            
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