**Using exponents with parentheses : **

When we evaluate expressions, we have to perform operations inside the parentheses.

"Evaluating Numerical Expressions" is one of the important basic stuff of algebra.

A numerical expression is an expression involving numbers and operations.You can use the order of operations to evaluate numerical expressions.

Order of operations can be done by using the following steps.

1. Perform operations in parentheses.

2. Find the value of numbers with exponents.

3. Multiply or divide from left to right.

4. Add or subtract from left to right.

**Example 1 :**

**Evaluate the following**

**Solution : **

Here we have 36 ÷ 12 inside the bracket, by dividing 36 and 12, we get 3.

= 6 x 3² + 8

Now we have to find the value of 3².

= 6 x 9 + 8

Now we have two signs multiplication and addition. According to the order of operation, we have to do multiplication and then addition.

= 54 + 8

= 62

**Example 2 :**

Evaluate the following

**Solution :**

Here we have (4 + 2)² in the inner most bracket. So, first we have to add 4 and 2 and find the square of the sum.

= 12 x (6² / 4) - 7

= 12 x (36 / 4) - 7

= 12 x 9 - 7

= 108 - 7

= 101

**Example 3 :**

Evaluate the following

**Solution :**

Here we have (11 - 9)³

in the inner most bracket. So, first we have to subtract 9 from 11 and then take cube for the result.

= 320 ÷ (2³ / 2) x 8

= 320 ÷ (8 / 2) x 8

= 320 ÷ 4 x 8

= 80 x 8

= 640

Let us see some more stuff about order of operation.

The rule or order that we use for evaluating numerical expressions in math is called "BODMAS" rule.

Very simply way to remember BODMAS rule!

**B -----> Brackets first (Parentheses)**

**O -----> Of (orders :Powers and radicals) **

**D -----> Division**

**M -----> Multiplication**

**A -----> Addition**

**S -----> Subtraction**

**Important notes :**

1. In a particular simplification, if you have both multiplication and division, do the operations one by one in the order from left to right.

2. Division does not always come before multiplication. We have to do one by one in the order from left to right.

3. In a particular simplification, if you have both addition and subtraction, do the operations one by one in the order from left to right.

Examples :

12 ÷ 3 x 5 = 4 x 5 = 20

13 - 5 + 9 = 8 + 9 = 17

**In the above simplification, we have both division and multiplication. From left to right, we have division first and multiplication next. So we do division first and multiplication next.**

**Example 4 :**

**Evaluate the expression ** 5 + 2 - 3

**Solution :**

In the above problem we have two signs addition and subtraction.

Step 1 :

First add the numbers which are in wither sides of positive sign.

= 5 + 2 - 3

Step 2 :

Now, we have to subtract 3 from 7.

= 7 - 3

= 4

**Example 5 :**

**Evaluate the expression **6(6 ÷ 2) x 9

**Solution :**

According to the rule BODMAS, first we have to consider the numbers which are in bracket or parenthesis.

= 6(6 ÷ 2) x 9

= 6 (3) x 9

Now, we have three numbers which are multiplying. So multiply those numbers

= 18 x 9 ==> 162

**Example 6 :**

**Evaluate **6(6) ÷ (2 x 9)

**Solution :**

According to the rule BODMAS, first we have to consider the numbers which are in bracket.

= 6(6) ÷ 18

= (6 x 6)/18

= 2

Now, we have three numbers which are multiplying. So multiply those numbers

= 18 x 9 ==> 162

**Example 7 :**

**Evaluate **100 ÷ (16 + 9) x 6

**Solution :**

According to the rule BODMAS, first we have to consider the numbers which are in bracket.

= 100 ÷ (16 + 9) x 6

= 100 ÷ 25 x 6

= 4 x 6 ==> 24

**Example 8 :**

**Evaluate the expression **(2 x 11 + 1) - (3 x 6 + 5)

**Solution :**

According to the rule BODMAS, first we have to consider the numbers which are in bracket.

= (2 x 11 + 1) - (3 x 6 + 5)

Inside the bracket, we have two signs multiplication and addition. First, we have to perform multiplication then we have to do addition.

= 23 - 23 ==> 0

- Generating equivalent numerical expressions
- Use repeated multiplication
- Division facts
- Exponents
- Using exponents
- Finding the value of a power
- Finding the value of each power
- Find the missing exponent
- Find the missing base
- Finding the factors of a number
- Finding the prime factorization of a number
- using ladder diagram for prime factorization
- Order of operations
- Exploring the order of operations
- Evaluating the numerical expression
- Using exponents with parentheses

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