 maths > exponents

Exponents : Precedence Order PEMA

what you'll learn...

overview

Redefining the precedence order in arithmetics with PEMA or BOMA.

This topic is repeated to make sure that it is clear to the students. The details are updated to explain handling exponents, roots, and logarithms.

recap

Numerical expressions are simplified with the following rules.

•  Precedence order BODMAS / PEMDAS

•  Left to Right sequence for same precedence

The sequence order specified is not desirable. If the sequence order is to be strictly followed, then some problems require unnecessary computation.

$3+4-3$$3 + 4 - 3$ has to be simplified to $=7-3=4$$= 7 - 3 = 4$, which involves two steps.

$4×7÷4$$4 \times 7 \div 4$ has to be simplified to $28÷4=7$$28 \div 4 = 7$, which involve two steps.

As part of studies on integers, the expressions involving subtraction are analyzed and it is understood that a subtraction is modified to addition of additive inverse of subtrahend. This was explained in the course on integers.

As part of studies on fractions, the expressions involving division are analyzed and it is understood that a division is modified to multiplication of multiplicative inverse of divisor. This was explained in the course on fractions.

better

Numerical expressions are simplified with a redefined rule.

•  Expression is simplified using precedence order PEMA. That is, Parentheses, Exponents, Multiplication, and Addition.

Note, the precedence order PEMA does not have roots and logarithms. The roots are equivalently exponents of fractions, and logarithms and exponents do not clash as we have two different notations ${\mathrm{log}}_{2}^{3}4$${\log}_{2}^{3} 4$ and ${\mathrm{log}}_{2}{4}^{3}$${\log}_{2} {4}^{3}$.

The precedence order PEMA does not have subtraction, as the subtractions are converted into addition of additive inverse. And, the precedence order PEMA does not have division, as the divisions are converted into multiplication of multiplicative inverse.

The advantage of this is explained in the simple example.

$3×4÷3$$3 \times 4 \div 3$

convert the division to multiplication.
$=3×4×\left(\frac{1}{3}\right)$$= 3 \times 4 \times \left(\frac{1}{3}\right)$

No need to follow the sequence order. Since all are multiplication, add any number to any other in the sequence that suits.
$=3×\left(\frac{1}{3}\right)×4$$= 3 \times \left(\frac{1}{3}\right) \times 4$

$=1×4$$= 1 \times 4$

example

Simplify $32÷11×44÷4$$32 \div 11 \times 44 \div 4$.

Note that it has $44$$44$ which equals product of $11$$11$ and $4$$4$.

This can be solved in two ways

First : $32÷11×44÷4$$32 \div 11 \times 44 \div 4$ $=\frac{32}{11}×44÷4$$= \frac{32}{11} \times 44 \div 4$ $=\frac{1408}{11}÷4$$= \frac{1408}{11} \div 4$ $=128÷4$$= 128 \div 4$ $=32$$= 32$

Second : $32÷11×44÷4$$32 \div 11 \times 44 \div 4$ $=32×\left(\frac{1}{11}\right)×44×\left(\frac{1}{4}\right)$$= 32 \times \left(\frac{1}{11}\right) \times 44 \times \left(\frac{1}{4}\right)$ $=32×\left(\frac{1}{11}\right)×11$$= 32 \times \left(\frac{1}{11}\right) \times 11$ $=32$$= 32$

Note that the flexibility to change the order of operation can be an advantage in calculations.

but why?

The reasons to redefine the precedence and sequence were explained in integers, fractions. The same is repeated as a quick review. It will take only a short time to review.

Students at 6th or 7th level may skip this explanation.

There are good reasons to adapt to PEMA.

When learning properties of numbers, we learn the commutative property of addition. $a+b=b+a$$a + b = b + a$. But the subtraction does not follow the commutative property. What this means is, the position of numbers cannot be modified to simplify the problem. The redefined precedence order PEMA converts the subtraction into addition and allows commutative property to be used to ones advantage. $a-b=a+\left(-b\right)=-b+a$$a - b = a + \left(- b\right) = - b + a$

Another property of numbers is associative property of addition. $a+\left(b+c\right)=\left(a+b\right)+c$$a + \left(b + c\right) = \left(a + b\right) + c$. But the subtraction does not follow the associative property. What this means is, the sequence of arithmetic operations cannot be modified to simplify the problem. The redefined precedence order PEMA converts the subtraction into addition and allows associative property to be used to ones advantage. $a-\left(b-c\right)=a+\left(\left(-b\right)+c\right)=\left(a+\left(-b\right)\right)+c$$a - \left(b - c\right) = a + \left(\left(- b\right) + c\right) = \left(a + \left(- b\right)\right) + c$.

algebraic why!

There are good reasons to adapt to PEMA.

Algebra is extensively based on the numerical expressions and the properties of numerical arithmetic. In an algebraic expression of multiple terms, with subtraction in some of them, simplification is possible only if the sequence order is not required to be followed.

For example, ${x}^{2}-x+2{x}^{2}-4-7x$${x}^{2} - x + 2 {x}^{2} - 4 - 7 x$.

This expression is intuitively understood to be ${x}^{2}+\left(-x\right)+2{x}^{2}+\left(-4\right)+\left(-7x\right)$${x}^{2} + \left(- x\right) + 2 {x}^{2} + \left(- 4\right) + \left(- 7 x\right)$.

Then it can be simplified into ${x}^{2}+2{x}^{2}+\left(-x\right)+\left(-7x\right)+\left(-4\right)$${x}^{2} + 2 {x}^{2} + \left(- x\right) + \left(- 7 x\right) + \left(- 4\right)$ which equals, $3{x}^{2}-8x-4$$3 {x}^{2} - 8 x - 4$.

divisive why!!

There are good reasons to adapt to PEMA.

When learning properties of numbers, we learn the commutative property of multiplication. $a×b=b×a$$a \times b = b \times a$. But the division does not follow the commutative property. What this means is, the order of arithmetic operations cannot be modified to simplify the problem. The redefined precedence order PEMA converts the division into multiplication and allows commutative property to be used to ones advantage. $a÷b=a×\left(\frac{1}{b}\right)=\frac{1}{b}×a$$a \div b = a \times \left(\frac{1}{b}\right) = \frac{1}{b} \times a$

Another property of numbers is associative property of multiplication. $a×\left(b×c\right)=\left(a×b\right)×c$$a \times \left(b \times c\right) = \left(a \times b\right) \times c$. But the division does not follow the associative property. What this means is, the sequence of arithmetic operations cannot be modified to simplify the problem. The redefined precedence order PEMA converts the division into multiplication and allows associative property to be used to ones advantage. $a÷\left(b÷c\right)$$a \div \left(b \div c\right)$$=a×\frac{1}{b×\left(1/c\right)}$$= a \times \frac{1}{b \times \left(1 / c\right)}$$=a×\left(\left(\frac{1}{b}\right)×c\right)$$= a \times \left(\left(\frac{1}{b}\right) \times c\right)$$=\left(a×\frac{1}{b}\right)×c$$= \left(a \times \frac{1}{b}\right) \times c$.

algebraically again!

There are good reasons in algebra to adapt to PEMA.

Algebra is extensively based on the numerical expressions and the properties of numerical arithmetic. In an algebraic expression of multiple factors, with division in some of them, simplification is possible only if the sequence order is not required to be followed.

For example, $\left(x+2\right)÷\left(x+4\right)×\left(x+4\right)×\left(x-8\right)÷\left(x+2\right)$$\left(x + 2\right) \div \left(x + 4\right) \times \left(x + 4\right) \times \left(x - 8\right) \div \left(x + 2\right)$.

This expression is intuitively understood to be $\left(x+2\right)×\frac{1}{\left(x+4\right)}×\left(x+4\right)×\left(x-8\right)×\frac{1}{\left(x+2\right)}$$\left(x + 2\right) \times \frac{1}{\left(x + 4\right)} \times \left(x + 4\right) \times \left(x - 8\right) \times \frac{1}{\left(x + 2\right)}$.

Then it can be simplified into $\left(x+2\right)×\frac{1}{\left(x+2\right)}×\frac{1}{\left(x+4\right)}×\left(x+4\right)×\left(x-8\right)$$\left(x + 2\right) \times \frac{1}{\left(x + 2\right)} \times \frac{1}{\left(x + 4\right)} \times \left(x + 4\right) \times \left(x - 8\right)$ which equals, $\left(x-8\right)$$\left(x - 8\right)$.

summary

Numerical Arithmetics Precedence Order : PEMA / BOMA
PEMA = Parentheses, Exponents, Multiplication and Addition

BOMA = Brackets, Order, Multiplication and Addition

Note : Division is handled as multiplication by multiplicative inverse of divisor. And, Subtraction is handled as addition of additive inverse of subtrahend.

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