Perform calculations using scientific notation

Scientific notation represents a quantity as $a \times 10^n$ , with $1 \leq |a| < 10$ , making place value and order of magnitude explicit. Calculations include multiplying and dividing by combining coefficient operations with the laws of integer exponents, and adding or subtracting by first expressing terms with a common power of ten, then renormalizing the result; estimation helps detect errors such as adding exponents during addition. This scope does not include complex numbers, logarithms, or generalized symbolic exponent manipulation beyond these numerical operations.

Detailed Explanation: Perform calculations using scientific notation

Scientific notation has the form

a \times 10^n,

where $1 \leq \vert a \vert < 10$ . To calculate:

Multiply: multiply the coefficients and add the exponents.

\left(a\times10^m\right)\left(b\times10^n\right)=(ab)\times10^{m+n}

Divide: divide the coefficients and subtract the exponents.

\frac{a\times10^m}{b\times10^n}=\frac{a}{b}\times10^{m-n}

Add or subtract: first rewrite the numbers so they have the same power of $10$ . Then add or subtract the coefficients. Do not add or subtract exponents.

Worked example

Calculate

\frac{4.5\times10^6+3.0\times10^5}{1.5\times10^2}.

Step 1: Rewrite the terms being added with the same power of $10$ .

Rewrite $3.0\times10^5$ using $10^6$ :

3.0\times10^5=0.30\times10^6.

So the numerator becomes

4.5\times10^6+0.30\times10^6.

Step 2: Add the coefficients.

4.5\times10^6+0.30\times10^6 =(4.5+0.30)\times10^6 =4.80\times10^6.

Step 3: Divide the coefficients and subtract the exponents.

\frac{4.80\times10^6}{1.5\times10^2} =\frac{4.80}{1.5}\times10^{6-2}.

=3.2\times10^4.

Therefore,

\boxed{3.2\times10^4}.

The answer is already in scientific notation because $3.2$ is between $1$ and $10$ . As a quick check, the numerator is about $4.8\times10^6$ , and dividing by about $10^2$ should give an answer around $10^4$ , which matches the result.

Learn by doing: Perform calculations using scientific notation

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Scientific Notation - Multiplying (1 Decimal Place)

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