When performing calculations that involve mixed operations such as multiplication, division, addition, and subtraction, it is essential to adhere to the order of operations. A helpful mnemonic for remembering this order is PENDES, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This structure ensures that calculations are performed correctly and consistently.

For example, consider the expression 1.89 \times 10^6 \times 3.005 and 5.21^3 \div (8.829 - 6.5 + 2.920). The first step is to evaluate any expressions within parentheses. In this case, we first multiply the numbers in brackets:

1.89 \times 10^6 \times 3.005 = 5679450. However, when reporting this result, we must consider significant figures. The number 1.89 \times 10^6 has 3 significant figures, while 3.005 has 4. Therefore, our final answer should also have 3 significant figures, resulting in 5.68 \times 10^6 when expressed in scientific notation.

Next, we calculate 5.21^3, which means multiplying 5.21 \times 5.21 \times 5.21. The initial result is 141.420761, but since all factors have 3 significant figures, we round this to 141.

Now, we address the subtraction and addition in the denominator: 8.829 - 6.5 + 2.920. The subtraction yields 2.329, which has 3 decimal places, while 2.920 has 3 decimal places as well. Therefore, the final result of this operation must be rounded to 1 decimal place, giving us 2.3.

Next, we add 2.3 + 2.920, which results in 5.220. Since 2.3 has 1 decimal place and 2.920 has 3, the final answer must also have 1 decimal place, resulting in 5.2.

Now, we multiply the results from the numerator and denominator. The multiplication of 5.68 \times 10^6 and 141 gives 8.0088 \times 10^8. Since both numbers have 3 significant figures, we round this to 8.01 \times 10^8.

Finally, we divide 8.01 \times 10^8 by 5.2. The result is 1.54 \times 10^8, but since 5.2 has 2 significant figures, we round our final answer to 1.5 \times 10^8.

In summary, when dealing with mixed operations, always follow the order of operations and apply the rules for significant figures: for multiplication and division, use the least number of significant figures; for addition and subtraction, use the least number of decimal places.