Operations with numbers and scientific notation

A positive exponent counts multiplication by 10: $10^4=10000$. A negative exponent counts division by 10: $10^{-4}=\frac{1}{10000}=0.0001$. The special value $10^0=1$ leaves a coefficient unchanged.

Increasing an exponent by 1 multiplies the value by 10, while decreasing it by 1 divides the value by 10. This relationship is the safest way to reason about decimal movement instead of memorizing an arrow rule without meaning.

The laws of indices remain valid: $10^m\times10^n=10^{m+n}$, $\frac{10^m}{10^n}=10^{m-n}$, and $(10^m)^r=10^{mr}$.

Place the decimal point after the first non-zero digit. Count how many places the decimal point moved from its original position. For a number greater than or equal to 10, the exponent is positive.

Keep every digit justified by the original value or stated precision. For example, $72800000=7.28\times10^7$. Writing $7.2800\times10^7$ would claim extra precision unless those trailing zeros were known to be significant.

For a positive value between 0 and 1, place the decimal after the first non-zero digit. Count the places moved to the right; the exponent is the negative of that count.

Leading zeros locate the decimal point but are not significant figures. In $0.0000520$, the digits $5$, $2$ and the final $0$ are significant, so the value is $5.20\times10^{-5}$.

Multiplying by $10^k$ shifts place value. If $k>0$, the value grows; if $k<0$, the value shrinks. Insert zeros when no digit occupies a required place.

A reliable check is to ask whether the ordinary number has the expected scale. For instance, $6.2\times10^{-5}$ must be positive and smaller than 1.

For positive normalized quantities, compare exponents first. A larger exponent gives a larger number. If the exponents match, compare coefficients.

A ratio answers how many times as large one quantity is. If $A=6\times10^8$ and $B=2\times10^5$, then $\frac{A}{B}=3\times10^3=3000$, so $A$ is 3000 times as large as $B$.

An order of magnitude describes power-of-ten scale. In an exam, state the comparison the question requests rather than treating exponent difference and exact ratio as interchangeable.

For multiplication, multiply coefficients and add exponents. For division, divide coefficients and subtract exponents. Normalize only after completing the operation.

Keep units with the calculation. Multiplying lengths can create square or cubic units; dividing distance by time creates a rate.

Coefficients may be added or subtracted only after the powers of ten match. Rewrite one term using the other exponent, combine coefficients, and normalize the result.

This is the scientific-notation version of aligning place values in ordinary addition. The invalid shortcut $(a\times10^m)+(b\times10^n)=(a+b)\times10^{m+n}$ should never be used.

Apply the exponent to both the coefficient and the power of ten. Then normalize and attach any powered unit.

For example, squaring a length in metres produces square metres: $(2.0\times10^3\text{ m})^2=4.0\times10^6\text{ m}^2$.

Significant figures communicate measurement precision. Start counting at the first non-zero digit. Zeros between non-zero digits are significant; leading zeros are not. A trailing zero after a decimal point is significant because it records measured precision.

Scientific notation makes intended precision clear. The quantities $3.2\times10^5$, $3.20\times10^5$, and $3.200\times10^5$ have two, three and four significant figures respectively, even though their displayed central values are equal.

Exact counts and defined conversions are not measured approximations. For example, 60 seconds in a minute is exact and should not force a calculated result to one significant figure.

Carry full calculator precision through intermediate steps and round only the final answer unless a question explicitly instructs otherwise. Retained extra digits are called guard digits and reduce accumulated rounding error.

If a question specifies a number of significant figures, obey it. If no accuracy is stated, give an appropriate level of precision for the data and context, commonly three significant figures for a non-exact decimal result.

Always check whether rounding changes the normalized coefficient. For example, rounding 9.996 to 3 significant figures gives 10.0, which must be renormalized to a coefficient below 10.

Most calculators use an $\text{EXP}$, $\text{EE}$, or $\times10^x$ key. To enter $6.02\times10^{23}$, type $6.02$, press that key, then enter $23$. Do not type an extra multiplication sign and another 10 unless your model requires it.

A display such as $4.7\text{E}{-8}$ means $4.7\times10^{-8}$. The display is convenient input or output notation, but it must be rewritten conventionally in your solution.

Use parentheses around compound numerators and denominators. Check the calculator mode if results appear automatically in scientific notation or ordinary notation; display mode changes appearance, not value.

Metric prefixes are powers of ten. For example, $1\text{ km}=10^3\text{ m}$, $1\text{ mm}=10^{-3}\text{ m}$, $1\,\mu\text{m}=10^{-6}\text{ m}$, and $1\text{ ns}=10^{-9}\text{ s}$.

Write a conversion factor equal to 1 so unwanted units cancel. This prevents an exponent from being reversed accidentally.

For area or volume, the conversion factor must also be squared or cubed. Since $1\text{ cm}=10^{-2}\text{ m}$, it follows that $1\text{ cm}^2=10^{-4}\text{ m}^2$, not $10^{-2}\text{ m}^2$.

Scientific notation appears in astronomy, microscopic measurement, chemistry, environmental science, computing, medicine, population data and finance. The method is the same, but interpretation depends on the units and the meaning of the variables.

In applications, start with a formula or relationship, substitute values with units, calculate using full precision, then state a contextual conclusion. A result of $2.3\times10^6$ means little until it is identified as bytes, dollars, cells or kilometres.

AI HL questions often test judgement as well as arithmetic: is the result physically possible, does it have the right order of magnitude, and is the chosen precision meaningful?

For a multi-step problem, identify the required quantity, list the data with units, choose relationships, perform the calculation without premature rounding, and validate the result. This structure makes long AI HL questions manageable.

Estimate the exponent before using a GDC. If values near a billion are divided by values near a million, the result should be near a thousand. An output near a trillion signals an input or exponent error.

Domain constraints can matter. Counts must be whole and non-negative; time and physical lengths are usually non-negative; a rounded estimate should not claim more precision than the inputs.

Write down the relationship before substitution. This shows mathematical communication and makes units and exponent operations easier to audit.

Use the GDC efficiently, but convert any $\text{E}$-display into $a\times10^k$. Normalize, attach units and round only at the end to the requested accuracy.

For compare, interpret or comment questions, include a sentence in context. For “show that” questions, show the transformation or calculation that reaches the stated result rather than quoting it.

If a final coefficient is outside the normalized interval, adjust the exponent. Then perform four checks: sign, coefficient, exponent and units.

The most common errors are leaving an unnormalized coefficient, giving the wrong sign to a small-number exponent, adding unequal powers directly, subtracting exponents in the wrong order, and copying $\text{E}$-notation into the final response.

Other frequent errors are rounding too early, counting leading zeros as significant, forgetting that a squared unit squares its conversion factor, and omitting units or contextual interpretation.

Repair a suspicious answer by returning to place value or by estimating with one-digit coefficients. A calculation involving only positive quantities cannot produce a negative result, and multiplication by a very small positive number should usually reduce scale.

Attempt each question before revealing the steps. The set progresses from notation and operations to AI HL application and validation.

Unless another accuracy is stated, keep exact values or give a sensible final accuracy supported by the data.

Use this checklist for quick revision. A strong answer combines correct calculation with correct notation, accuracy, units and interpretation.