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Squaring a number is an operation that involves multiplying the number by itself. In everyday mathematics, when we square a simple whole number, such as 4, we calculate it as
\(4^2 = 4 \times 4 = 16\).

However, when dealing with numbers in scientific notation, the process involves two separate parts: squaring the coefficient and squaring the power of ten. For example,
\( (2 \times 10^3)^2 = (2^2) \times (10^3)^2 = 4 \times 10^6 \).

It's important to approach the squaring of a number in scientific notation systematically to avoid errors. Each part of the notation needs to be squared separately and then recombined back into the standard form of scientific notation.

There are several exponent rules that make calculations with powers more manageable, and two vital rules that come into play when squaring numbers in scientific notation:

Multiplying with Same Base

When numbers with exponents have the same base, you can multiply them by adding the exponents. This is why in our example,
\(10^{-17} \times 10^{-17} = 10^{-17 + (-17)} = 10^{-34}\).

Power of a Power

When raising a power to another power, such as squaring, you multiply the exponents.
For the coefficient, it's just a standard multiplication, but for the power of ten, you'll apply the rule:
\((10^m)^n = 10^{m \times n}\).

Remembering these rules can simplify working with scientific notation immensely. They form the foundation of many operations you’ll encounter in algebra and higher-level mathematics.

When we multiply coefficients in scientific notation, we handle it just like any other decimal multiplication. If you are squaring a coefficient in scientific notation, you simply multiply the decimal number by itself. In the provided exercise, the coefficient is 1.02 and when squared:
\(1.02 \times 1.02 = 1.0404\).

Multiplying coefficients may seem straightforward, but precision is essential, especially as you may end up with a number that also needs to be adjusted into proper scientific notation form. If the result of your multiplication is not between 1 and 10, you’ll have to shift the decimal point to make it so, and adjust the exponent accordingly.