Multiply Square Roots: Calculate √5 × √10 Step-by-Step

Solve the following exercise:

$\sqrt{5}\cdot\sqrt{10}=$

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 terms in parentheses (in reverse direction):

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

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

$\sqrt{5}\cdot\sqrt{10}= \\ \downarrow\\ 5^{\frac{1}{2}}\cdot10^{\frac{1}{2}}=$

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

$5^{\frac{1}{2}}\cdot10^{\frac{1}{2}}= \\ (5\cdot10)^{\frac{1}{2}}=\\ 50^{\frac{1}{2}}=\\ \boxed{\sqrt{50}}$

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

Therefore, the correct answer is answer c.