August 8, 2024

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# Order of Operations with Exponents

The rule to evaluate  the value of a mathematical expression that includes  exponents [e.g. ( $3^{2}$  -6 ÷ 3)  x 8 ], is  PEMDAS Stands for  P: Parentheses, E: Exponents ,  M:Multiplication , D: Division,  A: Addition and S: Subtraction.

What do you do first in order of operations ?

The following steps must be followed to solve a mathematical expression that includes an exponents:

Step 1: Operations inside the parentheses should be solved first .

Step 2:  After performing operations inside of parenthesis , solve the exponents .

Step 3:  After the parentheses  and the exponents, perform multiplication or division from left to right depending on which operation comes first.

Step 4:  Finally, after multiplication and division,  perform addition and subtraction from left to right depending on which operation comes first.

## Order of Operations with Exponents(Solved examples)

(i) (9+43-4) -8-$3^{2}$

Sol. (9+43-4) -8-$3^{2}$=  48-8-$3^{2}$         (Performing the operations inside the parentheses)

= 48-8-9             (solving exponent)

=31                       (performing subtraction)

(ii)  33+10-$4^{2}$÷$2^{3}\times&space;31$

Sol. 33+10-$4^{2}$÷$2^{3}\times&space;31$= 33+10-16÷$8\times&space;31$                            (performing exponents)

=33+10-2$\times&space;31$                                        (performing division)

=33+10-62                                             (performing multiplication)

= -19                                                           (performing subtraction)

(iii)  70÷7-$3^{3}$

Sol. 70÷7-$3^{3}$=  70÷7-27                      (performing exponent)

=10-27                               (performing division)

=-17                                     (solving subtraction)

(iv) $\left&space;(&space;1^{4}\times&space;2^{2}+3^{3}\right&space;)-2^{5}$÷ 4

Sol. $\left&space;(&space;1^{4}\times&space;2^{2}+3^{3}\right&space;)-2^{5}$÷ 4 = (1 x 4+27) -32 ÷ 4

=( 4+27) -32 ÷ 4

=31-32  ÷ 4

= 31- 8

= 23

(v)  25-8 x 2 +$3^{2}$

Sol. 25-8 x 2 +$3^{2}$ =25-8 x 2 +9

= 25-16 +9

=9+9

=18

(vi) $\mathbf{9^{2}+3\times&space;\left&space;(&space;9-5&space;\right&space;)^{2}}$÷ 4

Sol.  $9^{2}+3\times&space;\left&space;(&space;9-5&space;\right&space;)^{2}$÷ 4 =$9^{2}+3\times&space;4^{2}$÷ 4

= 81  +3 x 16÷ 4

=81+48 ÷ 4

=81+12

=93

(vii) 36÷ $\mathbf{[20-(4\times&space;2)]+4^{2}-6}$

Sol.  36÷$[20-(4\times&space;2)]+4^{2}-6$= 36÷[20-8]$+4^{2}-6$                     (solving the operations inside the parentheses)

=36 ÷12$+4^{2}-6$

=36÷12+16-6                                 (solving exponent)

=3+16-6                                            (performing division)

=13

(viii) $\mathbf{\left&space;(&space;12+43-5&space;\right&space;)-2+4^{2}}$

Sol. $\left&space;(&space;12+43-5&space;\right&space;)-2+4^{2}$ =$\left&space;(&space;55-5&space;\right&space;)-2+4^{2}$

=$50-2+4^{2}$

=50-2+16

= 48+16

= 64

(ix) ($\mathbf{3\times&space;5^{2}}$ ÷ 15) $\mathbf{-(5-2^{2})}$

Sol. ($3\times&space;5^{2}$÷ 15) $-(5-2^{2})$=(3 x 25 ÷ 15) -(5-4)

=(75 ÷15)-(5-4)

= 5-4

=1

(x) $\boldsymbol{2\times6&space;}$ ÷$\boldsymbol{\left&space;(&space;8-2&space;\right&space;)-2^{3}+3&space;\times&space;4}$

Sol.  $2\times6$ ÷$\left&space;(&space;8-2&space;\right&space;)-2^{3}+3&space;\times&space;4$   =   2 x 6 ÷ 6 – $2^{3}$ +3 x 4

=  2 x 6 ÷ 6 – 8 +3 x 4

= 12 ÷ 6 – 8 +3 x 4

=2 – 8 +3 x 4

=2-8+12

=-6+12

=6

### Order of Operations with Exponents  (Unsolved examples with answers)

(i)  ( $3^{2}$  -6 ÷ 3)  x 8

(ii) $\left&space;(&space;6+6^{2}&space;\right&space;)\times&space;3$

(iii) $6^{2}-3\times&space;\left&space;(&space;3^{2}\times&space;2&space;\right&space;)+5$

(iv) $5^{3}-54+8&space;\times&space;3$

(v) $5+2^{3}\times$(22÷ 11)$-3^{2}\times&space;(4+5)$

(vi) 5+21÷ $3\times&space;2^{3}-61$

(vii) $7+2^{2}\times&space;6+2^{2}-6$

(viii) $2\times&space;5+3^{2}-96$ ÷ 4

(ix) 12 ÷4$\times&space;4^{2}\times&space;2^{2}$

(x) $5^{2}\times&space;5-67$

(xi) 33+10-$4^{2}$÷$2^{3}\times&space;31$

(xii) $6^{2}-24$÷ $2^{3}$  +14 x 3

(xiii) $9\times&space;9^{2}$÷  $3^{3}-&space;14$

(xiv)  $12+5\times&space;5^{3}$ ÷5-11

(xv) $2^{3}+45$ ÷ 9

Ans.  (i) 56    (ii) 126    (iii) -13   (iv) 95   (v) -60   (vi ) 0   (vii) 29  (viii) -5  (ix) 67  (x) 58  (xi) -19   (xii) 75   (xiii) 13

(xiv)  126    (xv) 13