**– The answer should be expressed as an integer that is rounded off to a proper number of significant figures.**

The aim of this article is to perform the **subtraction** of **two numbers** expressed in **exponential form**. The basic concept behind this article is the **Order of Operations**, the **PEMDAS Process**, and **Significant Figures**.

An **Operation** is a **mathematical process** such as **addition**, **subtraction**, **multiplication**, and **division** to solve an **equation**. **PEMDAS** **Rule** is the **sequence** in which these **operations** are performed. It is abbreviated as follows:

**“P”** represents the **Parenthesis (brackets)**.

**“E”** represents the **Exponents (Powers or Roots)**.

**“M & D”** represents the **Multiplication** and **Division** **Operations**.

**“A & S”** represents the **Addition** and **Subtraction** **Operations**.

**PEMDAS** rule defines that operations are to be solved starting from **Parenthesis (brackets)**, then **Exponents (Powers or Roots)**, then **Multiplication** and **Division** (from left to right), and lastly **Addition** and **Subtraction** (from left to right).

**Significant Figures** of a number are defined as the **number of digits** in the given number that are **reliable** and indicate the **accurate quantity**.

In solving equations, the following rules are used:

**(a)** For **Addition** and **subtraction** **operations**, the numbers are rounded by the **lowest number of decimals points**.

**(b)** For **Multiplication** and **division** **operations**, the numbers are rounded by the** lowest number of significant figures**.

**(c)** **Exponential** **terms** $n^x$ are only rounded by the **significant** **figures** in the **base of the exponent**.

## Expert Answer

Given numbers are:

\[a=4.659\times{10}^4\]

\[b=2.14\times{10}^4\]

We need to compute the number resulting from the **subtraction** of $a$ and $b$.

\[a-b=?\]

We will first analyze the **significant figures** of the **decimal numbers**. As per the **significant rule** for **addition** or **subtraction** of numbers having different **significant figures**, we will consider **rounding off** both numbers to the **lowest number of decimal points**.

$4.659$ has **three digits** after the** decimal point**.

$2.14$ has **two digits** after the **decimal point**.

Hence, we will **round off** $4.659$ till it has **two digits** after the **decimal point**:

\[a=4.66\times{10}^4\]

Now we will check the **significant figures** for **Exponential** **Terms.**

\[Exponential\ Term={10}^4\]

As for the** exponential terms**, the **number of significant figures** in the **base of the exponent** is considered. In both **exponential terms**, the **number of significant figures** in the **base of the exponent** is **two**.

Now that **significant figures** are sorted, we will solve the equation using the **PEMDAS Rule**.

\[a-b=4.66\times{10}^4-2.14\times{10}^4\]

Taking the **exponential term** common:

\[a-b=(4.66-2.14)\times{10}^4\]

As per the **PEMDAS Rule**, we will first solve the term in the **parenthesis (brackets)** as follows:

\[4.66-2.14=2.52\]

So:

\[a-b=2.52\times{10}^4\]

It can be expressed as follows:

\[{10}^4=10000\]

\[a-b=2.52\times 10000\]

\[a-b=25200\]

## Numerical Result

The result for the **subtraction** of given **two numbers** is:

\[4.659\times{10}^4-2.14\times{10}^4=2.52\times{10}^4\]

In **Integer form:**

\[4.659\times{10}^4-2.14\times{10}^4=25200\]

## Example

Compute the result of the given equation as per **PEMDAS Rule**.

\[58\div(4\times5)+3^2\]

**Solution**

As per **PEMDAS Rule**, we will **first** solve the **parenthesis**:

\[4\times5=20\]

\[58\div(4\times5)+3^2=58\div20+3^2\]

**Secondly**, we will solve the **exponent**:

\[3^2=9\]

\[58 \div 20+3^2=58 \div 20+9\]

**Thirdly**, we will solve **division**:

\[58 \div 20+9=2.9+9\]

**Finally**, we will solve the **addition**:

\[2.9+9=11.9\]

So:

\[58 \div (4\times 5)+3^2=11.9\]