Rule Statement

For a number x with a given number of significant figures, the result of x^n (where n is an exact integer) or the nth root of x should be rounded to the same number of significant figures as x. This rule applies because the operation is a form of repeated multiplication or division, and the relative error in the result is approximately n times the relative error in the base. For most practical purposes, especially when n is small (e.g., 2 or 3), the simple rule of preserving the number of significant figures is adequate.

Key Principle: The exponent or root index is treated as an exact number. Only the base’s precision limits the result.

This rule is consistent with the propagation of uncertainty for relative errors. According to the Guide to the Expression of Uncertainty in Measurement (GUM, JCGM 100:2008), the standard uncertainty of a power is approximately n times the relative uncertainty of the base. However, the significant figure rule is a simplified approximation that works well for typical engineering and scientific calculations.

Worked Examples

Example 1: Squaring a Measurement

Calculate (2.5)^2. The base 2.5 has two significant figures. The exact result is 6.25. Rounding to two significant figures gives 6.3 (using half-up rounding). Note that 6.25 rounds to 6.3 because the third digit is 5, and we round up the second digit from 2 to 3.

Example 2: Square Root

Calculate √4.0. The base 4.0 has two significant figures (the decimal point indicates the zero is significant). The exact square root is 2.0. Since 2.0 has two significant figures, the result is 2.0.

Example 3: Cube of a Number

Calculate (1.23)^3. The base has three significant figures. The exact result is 1.860867. Rounding to three significant figures gives 1.86.

Example 4: Cube Root

Calculate ∛8.0. The base 8.0 has two significant figures. The cube root is exactly 2.0 (since 2^3=8). So the result is 2.0.

Example 5: Higher Powers

Calculate (0.50)^4. The base has two significant figures. The exact result is 0.0625. In scientific notation, 6.25 × 10^-2. Rounding to two significant figures gives 6.3 × 10^-2 (or 0.063).

Counter-Examples

Common errors include:

• Using the exponent’s significant figures: For (2.5)^2, someone might think the exponent 2 has one sig fig, so the result should have one sig fig, giving 6. But the exponent is exact, not a measured quantity.
• Keeping too many digits: Reporting 6.25 for (2.5)^2 without rounding, implying more precision than the base.
• Rounding intermediate steps: For example, computing (2.5)^2 as 6.25, then rounding to 6.3, but then using 6.3 in further calculations. This is acceptable if done at the end, but rounding intermediate results can introduce errors.
• Applying the rule to the result of a root incorrectly: For √4.0, some might report 2 (one sig fig) because they think the root index 2 is the only significant number. But the base’s precision governs.

Common Mistakes

1. Confusing exact numbers with measured numbers: Exponents and root indices are exact; they have infinite significant figures.
2. Ignoring the base’s precision: Always look at the number under the exponent/root.
3. Using the rule for logarithms: The rule for logarithms is different (the number of decimal places in the result equals the number of significant figures in the argument). Do not mix them.
4. Forgetting to use scientific notation for very large or small results: This helps maintain the correct number of significant figures.

Quick Reference Table

Related Rules

This rule is an extension of the multiplication/division rule, since x^n is repeated multiplication. For addition/subtraction, the rule is based on decimal places. For logarithms, the rule is different: the number of decimal places in the result equals the number of significant figures in the argument. See our articles on Multiplication and Division and Logarithms for more details.

Standards Citation

While no standard explicitly states the significant figure rule for exponents and roots, the underlying principles are covered in:

• NIST TN 1297 (Section 7.2): Guidelines for expressing uncertainty, which recommends using significant figures consistent with the uncertainty.
• GUM (JCGM 100:2008) (Section 7.2.6): Rules for rounding results of measurements.
• ASTM E29: Standard practice for using significant digits in test data.
• ISO 80000-1: Quantities and units – general principles, which discusses rounding and significant figures.

These standards emphasize that the number of significant figures should reflect the measurement uncertainty. The simple rule for exponents and roots is a practical approximation that aligns with these guidelines for most cases.

Practice Problems

1. Calculate (3.45)^2 and round to the correct number of sig figs.
2. Find √9.0 and report with proper sig figs.
3. Compute (0.020)^3.
4. Evaluate ∛27.0.

Answers: 1) 11.9 (3 sig figs) – 3.45^2=11.9025, round to 3 sig figs = 11.9. 2) 3.0 (2 sig figs). 3) 8.0 × 10^-6 (2 sig figs) – 0.020^3 = 8.0e-6. 4) 3.0 (2 sig figs).

Sources & Further Reading

• NIST TN 1297: Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results.
• JCGM 100:2008: Evaluation of measurement data – Guide to the expression of uncertainty in measurement (GUM).
• ASTM E29-13: Standard Practice for Using Significant Digits in Test Data.
• ISO 80000-1:2009: Quantities and units – Part 1: General principles.

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