This question aims to find the values of **b** from the given equation by using **arithmetic laws**. The simple use of addition and multiplication with values within the brackets will give the value of b.

**Arithmetic** is the oldest branch of mathematics and the word arithmetic originated from the Greek word** “Arithmos,”** meaning number. This branch of mathematics deals with basic operations like **addition, multiplication, division, and subtraction**. It is the in-depth study of the laws and properties of these operations.

To solve these equations, we need to follow some order of applying operations. The **order of operation** is applying **brackets** first, then the operation of division. After** division**, apply **multiplication** and then **addition **and **subtraction**.

## Expert Answer

From the given equation:

\[ 3 ( 2b + 3 ) ^ { 2 } = 36 \]

\[ ( 2b + 3 ) ^ { 2 } = \frac { 36 }{ 3 } \]

\[ ( 2b + 3 ) ^ { 2 } = 12 \]

Taking square root on both sides:

\[ 2b + 3 = \pm \sqrt { 12 } \]

\[ 2b = \pm \sqrt { 12 } – 3 \]

Dividing the equation by 2:

\[ b = \frac { \pm 2\sqrt { 3 } – 3 } {2} \]

\[ b = \frac { – 3 + 2\sqrt { 3 }} {2} \]

\[ b = \frac { -3 – 2\sqrt { 3 }} {2} \]

**Numerical Results**

The values of b are $ b = \frac { – 3 + 2\sqrt { 3 }} {2} $ and $ b = \frac { -3 – 2\sqrt { 3 }} {2} $.

## Example

Find the value of b if the equation is $ 3 ( 4b + 3 ) ^ {2} = 9 $

From the given equation:

\[ 3 ( 4b + 3 ) ^ {2} = 9 \]

\[ ( 4b + 3 ) ^ {2} = \frac { 9 }{ 3 } \]

\[ ( 4b + 3 ) ^ {2} = 3 \]

Taking the square root on both sides:

\[ 4b + 3 = \pm \sqrt { 3 } \]

\[ 4b = \pm \sqrt { 3 } – 3 \]

Dividing the equation by 4:

\[ b = \frac { \pm \sqrt 3 – 3 } { 4 } \]

By rearranging the equation:

\[ b = \frac { – 3 + \sqrt 3 } { 4 } \]

\[ b = \frac { -3 – \sqrt 3 } { 2 } \]

For a simple equation:

\[ 2 ( 5b + 3 ) = 10 \]

\[ 10b + 6 = 10 \]

\[ 10b = 10 – 6 \]

\[ 10b = 4 \]

\[ b = \frac { 4 } { 10 } \]

\[ b = \frac { 2 } { 5 } \]

The value of b is $ b = \frac { 2 } { 5 } $.

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