EXPONENTS AND ORDER OF OPERATIONS

An expression like 53 is called a power. The number 5 is the base and 3 is the exponent.

53 = 5 ⋅ 5 ⋅ 5

In evaluating the power of 53, we have 3 factors of 5.

To evaluate an expression involving more than one operation, we agree to perform operations in the following order .

Order of Operations

1. Simplify the expressions inside grouping symbols, such as parentheses, brackets, and fraction bars.

2. Evaluate all powers.

3. Do all multiplications and divisions in order from left to right.

4. Do all additions and subtractions in order from left to right.

Example 1 :

Evaluate :

[(72 - 9) ÷ 8]2

Solution :

= [(72 - 9) ÷ 8]2

Evaluate power inside grouping symbols.

= [(49 - 9) ÷ 8]2

Evaluate expression inside grouping symbols.

= [40 ÷ 8]2

Divide 40 by 8.

= [5]2

Multiply 5 and 2.

= 10

Example 2 :

Evaluate :

{11 - 20 ÷ [(52 - 13)/3] + 8}  2

Solution :

= {11 - 20 ÷ [(52 - 13)/3] + 8}  2

Evaluate power inside grouping symbols.

= {11 - 20 ÷ [(25 - 13)/3] + 8}  2

Evaluate expression inside grouping symbols.

= {11 - 20 ÷ [12/3] + 8}  2

Evaluate expression inside grouping symbols.

= {11 - 20 ÷ 4 + 8}  2

Divide 20 by 4.

= {11 - 5 + 8}  2

Subtract 5 from 11.

= {6 + 8}  2

Evaluate expression inside grouping symbols.

= 14 ⋅ 2

Multiply.

= 28

Example 3 :

Evaluate :

19 - 3{20 - [(24 - 7)/4] x 8}

Solution :

= 19 - 3{20 - [(24 - 7)/4] x 8}

Evaluate power inside grouping symbols.

= 19 - 3{20 - [(16 - 7)/4] x 8}

Evaluate expression inside grouping symbols.

= 19 - 3{20 - [9/4] x 8}

Multiply 9/4 and 8.

= 19 - 3{20 - 72/4}

Divide 72 by 4.

= 19 - 3{20 - 18}

Evaluate expression inside grouping symbols.

= 19 - 3{2}

Multiply 3 and 2..

= 19 - 6

= 13

Example 4 :

Evaluate :

[72 ÷ 32 ⋅ 2]/6

Solution :

= [72 ÷ 32 ⋅ 2]/6

Evaluate power inside grouping symbols.

= [72 ÷ 9 ⋅ 2]/6

Divide 72 by 9.

= [8 ⋅ 2]/6

Multiply 8 and 2.

= 16/6

= 8/3

Example 5 :

Evaluate :

53 - (1/2)(12 + 12 ÷ 3)

Solution :

= 53 - (1/2)(12 + 12 ÷ 3)

Evaluate power.

= 125 - (1/2)(12 + 12 ÷ 3)

Evaluate expression inside grouping symbols (divide 12 by 3).

= 125 - (1/2)(12 + 4)

Evaluate expression inside grouping symbols.

= 125 - (1/2)(16)

Multiply 1/2 and 16.

= 125 - 16/2

Divide 16 by 2.

= 125 - 8

= 117

Example 6 :

Evaluate the following expression if a = -2, b = 3 and c = 5.

(2c/a)2 - 10[(a + b)/a)]

Solution :

= (2c/a)2 - 10[(a + b)/a)]

Substitute a = -2, b = 3 and c = 5.

= (2 ⋅ 5/(-2))2 - 10[(-2 + 3)/(-2)]

Evaluate expression inside grouping symbols.

= (10/(-2))2 - 10[(1/(-2)]

= (-5)2 - 10[-1/2]

Evaluate power.

= 25 - 10[-1/2]

Multiply 10 and -1/2.

= 25 + 10/2

= 25 + 5

= 30

Example 7 :

Evaluate the following expression if x = 4, y = 1 and z = -3.

9 - 2x ÷ (z + y)3

Solution :

= 9 - 2x ÷ (z + y)3

Substitute x = 4, y = 1 and z = -3.

= 9 - 2 ⋅ ÷ (-3 + 1)3

Evaluate expression inside grouping symbols.

= 9 - 2 ⋅ ÷ (-2)3

Evaluate power.

= 9 - 2 ⋅ ÷ (-8)

Multiply 2 and 4.

= 9 - 8/(-8)

Divide 8 by -8.

= 9 - (-1)

= 9 + 1

= 10

Example 8 :

Evaluate the following expression if a = 4, b = -3 and c = 7.

a3 - [(b2 + c)/a) + (ab + c)

Solution :

= a3 - [(b2 + c)/a) + (ab + c)

Substitute a = 4, b = -3 and c = 7.

= 43 - [((-3)2 + 7)/4) + (4(-3) + 7)

Evaluate 43, (-3)2 and 4(-3).

= 64 - [(9 + 7)/4)] + (-12 + 7)

Evaluate expression inside grouping symbols.

= 64 - [16/4] + (-5)

Divide 16 by 4.

= 64 - 4 + (-5)

= 64 - 4 - 5

= 60 - 5

= 55

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