Indices, roots, standard form and exact calculation Study Guide

Standard form is a way of writing very large or very small numbers compactly. A number is in standard form when it is written as $A \times 10^n$, where $A$ is a number between 1 and 10 (inclusive of 1, but exclusive of 10), and $n$ is an integer. For example, $3,000,000$ in standard form is $3 \times 10^6$, and $0.000005$ is $5 \times 10^{-6}$. The exponent $n$ tells you how many places the decimal point has moved. If $n$ is positive, the original number is large; if $n$ is negative, the original number is small. When performing calculations with numbers in standard form, it's often easiest to perform the operation on the 'A' parts and the powers of 10 separately. For multiplication, you multiply the 'A' parts and add the powers of 10. For division, you divide the 'A' parts and subtract the powers of 10. For addition and subtraction, it's usually best to convert the numbers to ordinary form, perform the calculation, and then convert the result back to standard form, or adjust one of the numbers so they both have the same power of 10 before adding/subtracting the 'A' parts. It is crucial that the final answer is always left in correct standard form, meaning the 'A' part must be between 1 and 10.

Exact calculations involve finding an answer without any rounding or approximation, ensuring the result is precisely correct. This is particularly important when working with numbers that cannot be expressed as terminating decimals, such as fractions like 1/3 or irrational numbers like √2 or π. When asked for an exact answer, you should leave your answer in terms of fractions, surds (square roots that don't simplify to a whole number), or multiples of constants like π, rather than converting them to decimals. For example, if a calculation results in 1/3, the exact answer is 1/3, not 0.33 or 0.333.... Similarly, if you calculate the diagonal of a square with side length 1, the exact answer is √2, not 1.414. Exact values maintain full precision, which is crucial in many mathematical contexts, especially when subsequent calculations depend on the initial result. Approximating too early can lead to 'rounding errors' that accumulate and make the final answer less accurate. Therefore, always preserve the exact form of numbers until any rounding is explicitly required at the very end of a problem.