Multiply Square Roots: √1 × √2 × √3 Solution Guide

In order to simplify the given expression, apply two laws of exponents:

a. The definition of root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

b. The law of exponents for exponents applied to terms in parentheses (in reverse order):

$x^n\cdot y^n =(x\cdot y)^n$

Begin by converting the square roots to exponents using the law of exponents mentioned in a':

$\sqrt{1}\cdot\sqrt{2}\cdot\sqrt{3} \\ \downarrow\\ 1^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\cdot3^{\frac{1}{2}}=$

Due to the fact that there is a multiplication operation between three terms with identical exponents, we are able to apply the law of exponents mentioned in b' (which also applies to multiplication of several terms in parentheses) Combine them together in a multiplication operation within parentheses that are also raised to the same exponent:

$1^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\cdot3^{\frac{1}{2}}= \\ (1\cdot2\cdot3)^{\frac{1}{2}}=\\ 6^{\frac{1}{2}}=\\ \boxed{\sqrt{6}}$

In the final steps, we performed the multiplication within the parentheses and once again used the definition of root as an exponent mentioned in a' (in reverse order) to return to root notation.

Therefore, the correct answer is answer d.