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Compute 4.659×10^4−2.14×10^4. Round the answer appropriately.

– 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 formThe 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 equationPEMDAS 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:

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

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