Graduate Management Admission Test (GMAT) Practice Test

How can the root of a number be found?

1. using a calculator

2. estimating

3. breaking the number down into prime factors

4. using logarithms

The correct answer is: breaking the number down into prime factors

Finding the root of a number can be approached through various methods, and breaking the number down into its prime factors is one effective technique, especially when manually calculating square roots or other roots of whole numbers. This method relies on the fundamental theorem of arithmetic, which states that every integer greater than 1 can be uniquely factored into prime numbers. When you break a number down into its prime factors, you can identify pairs (for square roots) or groups corresponding to the degree of the root being calculated. For example, if finding the square root of 36, which factors into \(2^2 \times 3^2\), you can take one of each pair of prime factors, leading to the result of 6. This approach is particularly useful for perfect squares or when simplifying expressions involving roots. Although other methods like using a calculator, estimating, or using logarithms can also yield results for finding roots, breaking down the number into prime factors provides a clear and systematic way to derive the root analytically and demonstrates a deeper understanding of number properties.