Order of Operations

To evaluate expressions with multiple operations, we need to perform these operations in the correct order. The order of operations can be remembered using the acronym BIDMAS:

Brackets, Indices, Division and Multiplication, Addition and Subtraction.

Let’s look at an expression:

2+3\times 4-3^{2}+25\div 5+(2^{2}-1)\times 2

To evaluate this expression, we’ll follow the BIDMAS rule:

1. Brackets: (2^{2}-1), which gives us 4-1=3

2. Indices: 3^{2}, which is 9

3. Division: 25\div 5, which is 5

4. Multiplication: 3\times 4=12 and 3\times 2=6

5. Addition: 2+12+5+6=25

6. Subtraction: 25-9=16

Therefore, 2+3\times 4+3^{2}+25\div 5+(2^{2}-1)\times 2=16

Let’s look at some more examples.

Examples

Example 1:

Evaluate the following expressions:

a) 36-(5+(3^{2}-2)\times 3)\div 13

b) (54\div 9\times 3-10)^{2}+15

a) 36-(5+(3^{2}-2)\times 3)\div 13=36-(5+7\times 3)\div 1336-(5+(3^{2}-2)\times 3)\div 13=36-(5+7\times 3)\div 13

=36-26\div 13=36-26\div 13

=36-2=36-2

=34=34

b) (54\div 9\times 3-10)^{2}+15=(6\times 3-10)^{2}+15(54\div 9\times 3-10)^{2}+15=(6\times 3-10)^{2}+15

=(18-10)^{2}+15=(18-10)^{2}+15

=8^{2}+15=8^{2}+15

=64+15=64+15

=79=79

Example 2:

Find the value of 6+8\times 9\div 3

6+8\times 9\div 3=6+8\times 36+8\times 9\div 3=6+8\times 3, since 9\div 3=39\div 3=3 is done first.

=6+24=6+24

=30=30

Example 3:

Find the value of \frac {1}{1+\frac {2}{3}}

First, 1+\frac {2}{3} =\frac {3}{3} +\frac {2}{3} =\frac {5}{3}1+\frac {2}{3} =\frac {3}{3} +\frac {2}{3} =\frac {5}{3}

Next, \frac{1}{1+\frac{2}{3}} = \frac{1}{\frac{5}{3}}\frac{1}{1+\frac{2}{3}} = \frac{1}{\frac{5}{3}}

To simplify, multiply the numerator and denominator by 3: \frac{1 \times 3}{\frac{5}{3} \times 3} = \frac{3}{5}\frac{1 \times 3}{\frac{5}{3} \times 3} = \frac{3}{5}

Example 4:

Evaluate the expression, giving your answer as simple as possible:

1+2^{2}-1\times 3+(4^{2}\div 2\times 3)

First, evaluate the power: 4^{2}=164^{2}=16

Next, perform the division and multiplication: 16\div 2\times 3=8\times 3=2416\div 2\times 3=8\times 3=24

Now, simplify the expression: 1+2^{2}-1\times 3+24=1+4-1\times 3+241+2^{2}-1\times 3+24=1+4-1\times 3+24

=1+4-3+24=1+4-3+24

=26=26

Example 5:

Evaluate { (2^{2}-3) }^{100}-({ 2^{3}-7) }^{200}

({ (2^{2}-3) }^{100}-({ 2^{3}-7) }^{200})=({ 1 }^{100}-({ 1 }^{200})({ (2^{2}-3) }^{100}-({ 2^{3}-7) }^{200})=({ 1 }^{100}-({ 1 }^{200})

=1-1=1-1

=0=0

Example 6:

When there are more brackets within brackets, perform the innermost brackets first and move outwards. For example, to evaluate:

2+5-(2^{2}+(6^{3}\div 12))^{2}\times 3

1. First, evaluate the innermost bracket: 6^{3}\div 12=216\div 12=186^{3}\div 12=216\div 12=18

2. Now, the expression becomes: 2+5-(2^{2}+18)^{2}\times 32+5-(2^{2}+18)^{2}\times 3

3. Evaluate the power: 2^{2}=42^{2}=4

4. Simplify the expression within the bracket: 4+18=224+18=22

5. The expression becomes: 2+5-(22)^{2}\times 32+5-(22)^{2}\times 3

6. Evaluate the power: (22)^{2}=484(22)^{2}=484

7. Finally, simplify the expression: 2+5-484\times 3=7-14522+5-484\times 3=7-1452

8. The result is: -1445-1445

Example 7:

Evaluate 3\times 5^{2}-12\div 4

1. First, evaluate the power: 5^{2}=255^{2}=25

2. Next, perform the division: 12\div 4=312\div 4=3

3. Now, simplify the expression with multiplication: 3\times 25=753\times 25=75

4. Finally, simplify the expression with subtraction: 75-3=7275-3=72

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