Skill: 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.