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 .

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 ⋅ 4 ÷ (-3 + 1)³

Evaluate expression inside grouping symbols.

= 9 - 2 ⋅ 4 ÷ (-2)³

Evaluate power.

= 9 - 2 ⋅ 4 ÷ (-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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