Solve: Product of Square Roots √6 × √2 × √3 × √1

Question

Solve the following exercise:

6231= \sqrt{6}\cdot\sqrt{2}\cdot\sqrt{3}\cdot\sqrt{1}=

Video Solution

Solution Steps

00:00 Simplify the following expression
00:03 The root of a number (A) multiplied by the root of another number (B)
00:07 equals the root of their product (A times B)
00:10 Apply this formula to our exercise, and convert to a single root
00:15 Calculate the products
00:22 Calculate the root of 36
00:25 This is the solution

Step-by-Step Solution

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

a. The definition of root as an exponent:

an=a1n \sqrt[n]{a}=a^{\frac{1}{n}}

b. The law of exponents for an exponent applied to a product in parentheses (in reverse direction):

xnyn=(xy)n 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:

6231=612212312112= \sqrt{6}\cdot\sqrt{2}\cdot\sqrt{3}\cdot\sqrt{1}= \\ \downarrow\\ 6^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\cdot3^{\frac{1}{2}}\cdot1^{\frac{1}{2}}=

Due to the fact that we have a multiplication operation of four 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:

612212312112=(6231)12=3612=36=6 6^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\cdot3^{\frac{1}{2}}\cdot1^{\frac{1}{2}}= \\ (6\cdot2\cdot3\cdot1)^{\frac{1}{2}}=\\ 36^{\frac{1}{2}}=\\ \sqrt{36}=\\ \boxed{6}

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 36.

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

Answer

6 6