maths > exponents

Exponents: Simplification of Expressions

what you'll learn...

overview

In this page, handling 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.

"express" means "say"

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

Consider $2+4+3$$2 + 4 + 3$. This can be evaluated to $9$$9$. This is an example of a numerical expression.

The word "expression" means: collection of numbers and arithmetic operations between them, which together represent a quantity.

$2×4×3$$2 \times 4 \times 3$ is an example of a numerical expression. It is evaluated to $24$$24$.

$a+3x$$a + 3 x$ is not a numerical expression. It is not entirely numbers and arithmetic operations. It has letters a and x.

Is $3+4×2-6÷3$$3 + 4 \times 2 - 6 \div 3$ a numerical expression?
"Yes, all arithmetic operations can be part of a numerical expression"

what order

Consider $1+2$$1 + 2$ and $3×1$$3 \times 1$. Note that when evaluated, both result in identical numerical value $1+2=3$$1 + 2 = 3$ and $3×1=3$$3 \times 1 = 3$.

They are two different expressions, evaluating to equal values.

Consider $9-6÷3$$9 - 6 \div 3$.

To simplify the expression, the division is performed first. It is

$9-6÷3$$9 - 6 \div 3$

$=9-2$$= 9 - 2$

$=7$$= 7$

Division has higher precedence over subtraction.

In a numerical expression the precedence order is given as :

•  Brackets / Parentheses

•  Order / Exponent

•  Division and Multiplication

This is abbreviated as BODMAS or PEMDAS.

what sequence?

Consider $20-4-3$$20 - 4 - 3$.

It is wrong to do $20-4-3$$20 - 4 - 3$ $\ne 20-1$$\ne 20 - 1$ $=19$$= 19$.

The correct order of simplification is $20-4-3$$20 - 4 - 3$ $=16-3$$= 16 - 3$ $=13$$= 13$

The two subtractions are in the same precedence level. This is to be handled from left to right sequence.

Consider $36÷6÷3$$36 \div 6 \div 3$

It is wrong to simplify as $36÷6÷3$$36 \div 6 \div 3$ $\ne 36÷2$$\ne 36 \div 2$ $=18$$= 18$

The correct order of simplification is $36÷6÷3$$36 \div 6 \div 3$ $=6÷3$$= 6 \div 3$ $=2$$= 2$.

The two divisions are in same precedence level. This is to be handled from left to right sequence. .

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

out of order / sequence

Consider $6÷3×2$$6 \div 3 \times 2$.

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

$6÷3×2$$6 \div 3 \times 2$

$=2×2$$= 2 \times 2$

$=4$$= 4$

Consider $6÷\left(3×2\right)$$6 \div \left(3 \times 2\right)$

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

$6÷\left(3×2\right)$$6 \div \left(3 \times 2\right)$

$=6÷6$$= 6 \div 6$

$=1$$= 1$

The rule of brackets or parentheses in numerical expression is, the subexpression within a bracket or parentheses has the highest precedence

everything needed

In a numerical expression, the precedence order is:
•  Parentheses or Brackets are at the highest precedence order

•  division and multiplication in same level same level of precedence
•  addition and subtraction in same level of precedence.

This is abbreviated as BODMAS (Division, Multiplication, Addition, Subtraction) or PEMDAS (Multiplication, Division, Addition, Subtraction).

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

All these were studied as part of whole numbers and integers. The same applies for fractions.

•  Precedence order BODMAS / PEMDAS

•  Left to Right sequence for same precedence

Let us modify this to include exponents.

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.

examples

Simplify $11-{3}^{2}$$11 - {3}^{2}$
The answer is "$2$$2$".

The exponent is of higher precedence than subtraction.

$11-{3}^{2}$$11 - {3}^{2}$

$=11-9$$= 11 - 9$

$=2$$= 2$

Simplify ${\mathrm{log}}_{3}9+18$${\log}_{3} 9 + 18$
The answer is "$20$$20$".

The logarithm is of higher precedence than addition.

${\mathrm{log}}_{3}9+18$${\log}_{3} 9 + 18$

$=2+18$$= 2 + 18$

$=20$$= 20$

Simplify $4×{4}^{\frac{1}{2}}$$4 \times {4}^{\frac{1}{2}}$
The answer is "$8$$8$".

The square root is of higher precedence than subtraction.

$4×{4}^{\frac{1}{2}}$$4 \times {4}^{\frac{1}{2}}$

$=4×2$$= 4 \times 2$

$=8$$= 8$

Simplify ${\mathrm{log}}_{2}{4}^{3}$${\log}_{2} {4}^{3}$

Which one of the following is correct?

${\mathrm{log}}_{2}{4}^{3}$$\textcolor{c \mathmr{and} a l}{{\log}_{2} {4}^{3}}$
$={2}^{3}$$\textcolor{c \mathmr{and} a l}{= {2}^{3}}$
$=8$$\textcolor{c \mathmr{and} a l}{= 8}$

${\mathrm{log}}_{2}{4}^{3}$$\textcolor{\mathrm{de} e p s k y b l u e}{{\log}_{2} {4}^{3}}$
$={\mathrm{log}}_{2}\left({2}^{6}\right)$$\textcolor{\mathrm{de} e p s k y b l u e}{= {\log}_{2} \left({2}^{6}\right)}$
$=6$$\textcolor{\mathrm{de} e p s k y b l u e}{= 6}$

The answer is "$6$$6$".

The notation of cube of log is ${\mathrm{log}}_{2}^{3}4$${\log}_{2}^{3} 4$. The notation for log of a cube is ${\mathrm{log}}_{2}{4}^{3}$${\log}_{2} {4}^{3}$.

${\mathrm{log}}_{2}^{3}4$$\textcolor{c \mathmr{and} a l}{{\log}_{2}^{3} 4}$
logarithm is applied first
$={2}^{3}$$\textcolor{c \mathmr{and} a l}{= {2}^{3}}$
$=8$$\textcolor{c \mathmr{and} a l}{= 8}$

${\mathrm{log}}_{2}{4}^{3}$$\textcolor{\mathrm{de} e p s k y b l u e}{{\log}_{2} {4}^{3}}$
exponent is applied first
$={\mathrm{log}}_{2}\left({2}^{6}\right)$$\textcolor{c \mathmr{and} a l}{= {\log}_{2} \left({2}^{6}\right)}$
$=6$$\textcolor{c \mathmr{and} a l}{= 6}$

Simplify $4-6÷{3}^{-2}$$4 - 6 \div {3}^{- 2}$
The answer is "$-50$$- 50$".

$4-6÷{3}^{-2}$$4 - 6 \div {3}^{- 2}$

exponent is of higher precedence
$=4-6÷\left(\frac{1}{9}\right)$$= 4 - 6 \div \left(\frac{1}{9}\right)$

division is of higher precedence
$=4-54$$= 4 - 54$

$=-50$$= - 50$.

Simplify ${\mathrm{log}}_{10}100+9900$${\log}_{10} 100 + 9900$.
The answer is "$9902$$9902$".

${\mathrm{log}}_{10}100+9900$${\log}_{10} 100 + 9900$

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

$=9902$$= 9902$

Simplify $\sqrt[3]{\left(-1+3-1\right)}+\left(-2\right)-1$$\sqrt[3]{\left(- 1 + 3 - 1\right)} + \left(- 2\right) - 1$
The answer is "$-2$$- 2$"

$\sqrt[3]{\left(-1+3-1\right)}+\left(-2\right)-1$$\sqrt[3]{\left(- 1 + 3 - 1\right)} + \left(- 2\right) - 1$

Bracket is of the highest precedence
$=\sqrt[3]{1}+\left(-2\right)-1$$= \sqrt[3]{1} + \left(- 2\right) - 1$

exponent or root is higher in precedence
$=1+\left(-2\right)-1$$= 1 + \left(- 2\right) - 1$

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

$=-2$$= - 2$

summary

Simplification of Expressions : BODMAS

•  B - Brackets

•  O - Order (exponents, roots, logarithm)

•  D - Division

•  M - Multiplication

•  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

•  S - Subtraction

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

Outline

The outline of material to learn "Exponents" is as follows. Note: click here for detailed outline of Exponents s

→   Representation of Exponents

→   Inverse of exponent : root

→   Inverse of exponent : Logarithm

→   Common and Natural Logarithms

→   Exponents Arithmetics

→   Logarithm Arithmetics

→   Formulas

→   Numerical Expressions

→   PEMA / BOMA

→   Squares and Square roots

→   Cubes and Cube roots