Graduate Management Admission Test (GMAT) Practice Test

What is the product of two n-th roots expressed as (ⁿ√x)(ⁿ√y)?

1. ⁿ√(x + y)

2. ⁿ√(xy)

3. (xy)ˡ/ⁿ

4. (x*y)ⁿ

The correct answer is: ⁿ√(xy)

The product of two n-th roots, expressed as (ⁿ√x)(ⁿ√y), can be simplified using the properties of exponents and roots. When you multiply two n-th roots, you are essentially combining the radicands (the numbers under the root) before taking the n-th root of the result. To break it down mathematically: 1. Each n-th root can be expressed using exponents: - The n-th root of \(x\) is equivalent to \(x^{1/n}\). - The n-th root of \(y\) is equivalent to \(y^{1/n}\). 2. When you multiply these two roots together, you have: - \(ⁿ√x * ⁿ√y = x^{1/n} * y^{1/n}\). 3. According to the rules of exponents, when you multiply like bases you add the exponents: - This can be rewritten as \( (xy)^{1/n} \). 4. Finally, we recognize that this is equivalent to taking the n-th root of the product of \(x\) and \(y\): - Thus, \(ⁿ√x * ⁿ√y =