EQN-06-06

Comparison with Exponents & Roots

Compare powers and roots by matching bases or exponents — and beware the fraction-and-negative traps.

~5 min read•Difficulty 3/5

Match the base or the exponent

If you can write both quantities with the same base, the larger exponent wins (for a base $> 1) .$

A : 2^{10} B : 4^{4} = 2^{8} \to A is greater

If you can match the exponent, the larger base wins.

Powers of numbers between 0 and 1

A fraction between 0 and 1 gets smaller as the exponent grows.

(1/2)^{2} = 1/4 < 1/2 \to higher power is smaller here

Negatives flip intuition

A negative base raised to an even power is positive; to an odd power, negative. Never assume a higher power means a larger value when the base could be negative.

Roots shrink large numbers, grow small ones

For $n > 1, n < n$ ; for $0 < n < 1, n > n .$ Comparisons that mix a number and its root hinge on whether the number is above or below 1.

Worked examples

Example 1 (difficulty 2/5)

Quantity

A : 3^{5.}

Quantity

B : 9^{2.}

Which is greater?

◎ Predict the answer first:

Example 2 (difficulty 3/5)

Given

0 < x < 1.

Quantity A: x². Quantity B: x³. Which is greater?

◎ Predict the answer first:

Tips

Rewrite both quantities with a common base when possible, then compare exponents.
For bases between 0 and 1, the higher power is the SMALLER number.
If the base could be negative, check even vs odd power before deciding.

Common mistakes

Assuming a higher exponent always means a larger value

Why: True only for bases greater than 1.

Fix: For

0 <

base

< 1,

higher powers shrink; for negatives, signs alternate.

Comparing powers with different bases without converting

Why: The exponents look directly comparable.

Fix: Match the base (or exponent) first, or estimate both values.

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