Standard form writes any number as $A \times 10^n$, where $1 \le A < 10$ and $n$ is an integer. It's used so very large or very small numbers can be written, compared, and calculated with more easily. The coefficient $A$ carries the significant digits; the power $n$ tells you the magnitude and which way to shift the decimal point.

Place the decimal point immediately after the first non-zero digit, then count how many places it moved:

• Large numbers → positive $n$. $37{,}000 \to 3.7\times10^4$ (decimal moved 4 places left).
• Small decimals → negative $n$. $0.000045 \to 4.5\times10^{-5}$ (decimal moved 5 places right; the number is smaller than $4.5$, so $n$ must be negative).

Separate the coefficients from the powers of ten, then combine:

Example: $(3\times10^4)\times(6\times10^5)$: coefficients $3\times6=18$; powers $4+5=9$; preliminary result $18\times10^9$. Since $18\ge10$, divide it by 10 (giving $1.8$) and add 1 to the power: $1.8\times10^{10}$.

For division, subtract the powers instead of adding them: $10^{n_1}\div10^{n_2}=10^{n_1-n_2}$.

The index laws don't apply to addition/subtraction. Instead, rewrite both numbers with the same power of ten (usually the larger one), then add/subtract the coefficients directly.

Example: $(2.4\times10^5)+(1.68\times10^2)$. Rewrite the second term with power $10^5$: $1.68\times10^2 = 0.00168\times10^5$. Then $2.4+0.00168 = 2.40168$, giving $2.40168\times10^5$.

This "align the powers" method is essential once the exponents are far apart — converting everything to ordinary numbers first becomes impractical.

Divide one by the other, handling coefficients and powers separately. If Jupiter's surface area is $6.22\times10^{15}\text{ km}^2$ and Earth's is $5.10\times10^{14}\text{ km}^2$:

Worked Example

Given $x = 8.5\times10^9$ and $y = 4\times10^8$, find $x+y$ in standard form.

1. Align powers: rewrite $y$ with power $10^9$: $4\times10^8 = 0.4\times10^9$.
2. Add coefficients: $8.5+0.4 = 8.9$.
3. Result: $8.9\times10^9$ (already satisfies $1\le A<10$, so no further adjustment needed).

For small exponents this works fine, but it quickly becomes impractical — and in a non-calculator question, essentially impossible — once the powers involved are large (e.g. $10^{15}$ or $10^{-8}$). The "align the powers" method for addition/subtraction, and the "separate coefficients from powers" method for multiplication/division, exist precisely because they let you work with the compact standard-form numbers directly, without ever writing out a string of zeros. Treating standard form as a genuine calculation tool — rather than just a display format to convert in and out of — is what the exam is really testing.