To multiply numbers in standard form, follow these steps: multiply the coefficients and add the exponents of the powers of ten.

First, identify the coefficients (the numbers in front of the powers of ten) and the exponents (the powers of ten themselves). For instance, if you have two numbers in standard form, represented as $a \times 10^m$ and $b \times 10^n$, you will multiply the coefficients $a$ and $b$ together. Subsequently, you will add the exponents $m$ and $n$.

Let’s illustrate this process with an example: multiplying $3 \times 10^4$ by $2 \times 10^3$. Start by multiplying the coefficients:

Next, add the exponents:

Thus, the product is:

If the product of the coefficients results in a number that falls outside the range of 1 to 10, you will need to adjust it to maintain standard form. For example, consider multiplying $4 \times 10^2$ by $5 \times 10^3$. Here, you compute:

And for the exponents:

Since $20$ is not between 1 and 10, you must convert it to standard form. This involves writing $20$ as $2 \times 10^1$ and then multiplying by $10^5$:

In summary, remember these key steps: multiply the coefficients, add the exponents, and adjust the result to ensure the coefficient is between 1 and 10. This method will keep your calculations organized and ensure that your answers are presented in the correct standard form.