Multiply Square Roots: Solving √4 × √2 × √2 Step-by-Step

Solve the following exercise:

$\sqrt{4}\cdot\sqrt{2}\cdot\sqrt{2}=$

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 an exponent applied to a product in parentheses (in reverse direction):

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

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

$\sqrt{4}\cdot\sqrt{2}\cdot\sqrt{2} \\ \downarrow\\ 4^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\cdot2^{\frac{1}{2}}=$

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

$4^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\cdot2^{\frac{1}{2}}= \\ (4\cdot2\cdot2)^{\frac{1}{2}}=\\ 16^{\frac{1}{2}}=\\ \sqrt{16}=\\ \boxed{4}$

In the final steps, we first performed the multiplication within the parentheses, we then once again used the definition of root as an exponent mentioned in a' (in reverse direction) to return to root notation, and in the final stage, we calculated the known square root of 16.

Therefore, we can identify that the correct answer is answer c.