Reasoning vs Computation Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy
Without long division, determine whether 2102^{10} is greater or less than 10310^3. Explain your reasoning.

Solution

  1. 1
    210=10242^{10} = 1024 and 103=100010^3 = 1000.
  2. 2
    Reasoning: 210=(25)2=322=1024>10002^{10} = (2^5)^2 = 32^2 = 1024 > 1000.

Answer

210=1024>1000=1032^{10} = 1024 > 1000 = 10^3
Knowing 210=10242^{10} = 1024 (a standard fact) gives the answer immediately. Alternatively, 210>1032^{10} > 10^3 because 210/103=1024/1000>12^{10}/10^3 = 1024/1000 > 1. Structural knowledge beats repeated multiplication.

About Reasoning vs Computation

Reasoning is the process of understanding why a mathematical fact is true and how ideas connect, while computation is the mechanical process of calculating an answer — both are essential but serve different purposes.

Learn more about Reasoning vs Computation →

More Reasoning vs Computation Examples

Example 1 easy

Without computing, determine whether [formula] is greater than, less than, or equal to [formula]. Th

Example 2 medium

Use reasoning (not direct computation) to find the units digit of [formula].

Example 4 medium

By reasoning (not expanding), determine the degree of the polynomial [formula].

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