Multiply Square Roots: Calculate √2 × √3

Question

Solve the following exercise:

23= \sqrt{2}\cdot\sqrt{3}=

Video Solution

Solution Steps

00:00 Solve the following problem
00:03 The square root of a number (A) multiplied by the square root of another number (B)
00:07 equals the square root of their product (A times B)
00:11 Apply this formula to our exercise and proceed to calculate the product
00:14 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 exponents applied to terms in parentheses (in reverse):

xnyn=(xy)n x^n\cdot y^n =(x\cdot y)^n

Let's start by converting the square roots to exponents using the law of exponents mentioned in a':

23=212312= \sqrt{2}\cdot\sqrt{3}= \\ \downarrow\\ 2^{\frac{1}{2}}\cdot3^{\frac{1}{2}}=

Due to the fact that there is multiplication between two terms with identical exponents, we can apply the law of exponents mentioned in b' and then proceed to combine them together inside of parentheses, which are raised to the same exponent:

212312=(23)12=612=6 2^{\frac{1}{2}}\cdot3^{\frac{1}{2}}= \\ (2\cdot3)^{\frac{1}{2}}=\\ 6^{\frac{1}{2}}=\\ \boxed{\sqrt{6}}

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

Therefore, the correct answer is answer b.

Answer

6 \sqrt{6}