Can you explain how to simplify 10^-(pH1 - 3) to 10^-pH1 x 103?

$$10^{-(pH_1 - 3)}$$

When you distribute the negative sign, it changes the sign of each term inside the parentheses:

$$-(pH_1 - 3) = -pH_1 - (-3) = -pH_1 + 3$$

So, the expression becomes:

$$10^{-pH_1 + 3}$$

In our case, $a=10$, $m=-pH_1$, and $n=3$.

$$10^{-pH_1 + 3} = 10^{-pH_1} \times 10^3$$

This is a fundamental rule of exponents: when you multiply powers with the same base, you add their exponents. Conversely, an exponent that is a sum can be split into a product of powers.

So, the full transformation is:

$$[H_3O^+]_2 = 10^{-(pH_1 - 3)}$$ $$[H_3O^+]_2 = 10^{-pH_1 + 3}$$ $$[H_3O^+]_2 = 10^{-pH_1} \times 10^3$$