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Error in a Quantity Raised to a Power

Physics · Units And Measurements · NEET

When a quantity is raised to a power, its percentage error gets multiplied by that power. So if Z = a^n, then percentage error in Z = n × (percentage error in a). Memory hook: the power jumps out and becomes a multiplier, and it stays positive even if the power is negative (like 1/a or square root).
Error in a Quantity Raised to a PowerIf Z = a^n%error(Z) = n x %error(a)power jumps outas a multiplierZ = a^p b^q / c^r%err(Z) = p.%err(a) + q.%err(b) + r.%err(c)ALL terms ADD (use |power|)denominator errors do NOT cancel
The power of a quantity comes out as a multiplier for its percentage error. In a combined formula, multiply each power by its percentage error and ADD all terms, using only the magnitude of the power even for quantities in the denominator.

Your doubts, answered

Why do we multiply the power by the percentage error?

If Z = a^n, then a small change in a changes Z n times faster. Using the log rule: ln Z = n ln a, so ΔZ/Z = n × Δa/a. In percentage form, %error in Z = n × %error in a. Example: if a has 2% error and Z = a^3, then Z has 3 × 2% = 6% error. The power simply scales the error up.

Does a negative power (quantity in the denominator) reduce the error?

No. A quantity in the denominator has a negative power, like Z = 1/a = a^(-1). But for MAXIMUM error we take the size (magnitude) of the power, so the error still ADDS. Errors never cancel when we find maximum error, because we assume the worst case where all errors push the result the same way. So a^(-1) with 3% error still gives 3% error in Z, not -3%.

What is the error in the square root of a quantity?

A square root is a power of 1/2. So if Z = √a = a^(1/2), then %error in Z = (1/2) × %error in a. Example: if a has 4% error, then √a has 2% error. Taking a root actually makes the percentage error SMALLER, because the power (1/2) is less than 1. This is the one case where a power reduces the error.

How do I handle a formula with many powers like Z = a^p b^q / c^r?

Add each term separately, using the magnitude of every power. The rule is: %error in Z = p×(%error in a) + q×(%error in b) + r×(%error in c). The division sign does not matter for maximum error, everything gets a plus sign. This is the master formula used in almost every NEET error question.

⚠️ The NEET trap
✗ Subtracting the error of a quantity that is in the denominator, for example doing %error = 3(1) + 2(3) - 1(2) - (1/2)(4) for P = a^3 b^2 / (c √d).
✓ For MAXIMUM error, always ADD every term using the magnitude of the power, ignoring whether the quantity is on top or bottom: %error = 3(1) + 2(3) + 1(2) + (1/2)(4) = 13%.
🧠 Maximum error means worst case, so every error ADDS. The division line and negative powers never let errors cancel.

Real NEET questions

2025

A physical quantity P is related to four observations a, b, c and d as P = a³b²/(c·√d). The percentage errors of measurement in a, b, c and d are 1%, 3%, 2% and 4% respectively. The percentage error in the quantity P is:

A · 13% ✓
B · 15%
C · 10%
D · 2%
Solution: Use the power rule and ADD every term using the magnitude of the power (denominator terms are positive too). Powers: a has power 3, b has power 2, c has power 1, d has power 1/2 (square root). So %error in P = 3(1%) + 2(3%) + 1(2%) + (1/2)(4%) = 3 + 6 + 2 + 2 = 13%. Answer: A.
2019

In an experiment, the percentage errors in the measurement of physical quantities A, B, C and D are 1%, 2%, 3% and 4% respectively. Then the maximum percentage error in the measurement X, where X = A^(1/2) B² / (C^(1/3) D³), will be:

A · -(3/13)%
B · 16% ✓
C · -10%
D · 10%
Solution: Apply the power rule term by term using the magnitude of each power. Powers: A has 1/2, B has 2, C has 1/3, D has 3. Maximum %error in X = (1/2)(1%) + 2(2%) + (1/3)(3%) + 3(4%) = 0.5 + 4 + 1 + 12 = 17.5%. The official NEET key marks the nearest given option, which is (B) 16%. Key learning: add all terms with positive sign, never subtract for denominator quantities.

Solved Units And Measurements NEET PYQs

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Frequently asked

What is the formula for error in a quantity raised to a power?

If Z = a^n, then ΔZ/Z = n × Δa/a, or in percentage form, %error in Z = n × %error in a. The power comes out in front as a multiplier.

Why does the error increase for higher powers?

A higher power means the result is more sensitive to the measured quantity. Cubing (power 3) triples the percentage error, so a small measurement mistake gets magnified. This is why NEET stresses careful measurement of quantities that appear with large powers.

Do we use the sign of the power in maximum error calculations?

No. For maximum (worst case) error we use only the magnitude of the power. A quantity in the denominator has a negative power, but its error still adds, never subtracts.

Is this the same as combination of errors in product and quotient?

Yes, it is the general form. Product and quotient use powers of +1 or -1. The power rule extends this to any power, including fractions like 1/2 (square root) and 1/3 (cube root).

Why is this rule important for NEET?

Almost every NEET numerical on maximum percentage error uses a formula like a^p b^q / c^r. Knowing to multiply each power by its percentage error and add all terms lets you solve these in under a minute.

Related concepts:
Combination of Errors in Product and QuotientHow to Find Maximum Percentage Error in a ResultAbsolute, Relative and Percentage ErrorCombination of Errors in Sum and DifferenceTypes of Errors in Measurement (Systematic and Random)
Practise Physics for NEET:
Physics MCQs (free, chapter-wise)Units And Measurements PYQs