Multiply Square Roots: Calculate √2 × √5 Step-by-Step

Question

Solve the following exercise:

25= \sqrt{2}\cdot\sqrt{5}=

Video Solution

Solution Steps

00:00 Solve
00:03 Square root of a number (A) times square root of another number (B)
00:07 Equals the square root of their product (A times B)
00:11 We'll use this formula in our exercise and calculate the product
00:14 And this is the solution to the question

Step-by-Step Solution

In order to simplify the given expression we use two laws of exponents:

A. Defining the root as an exponent:

an=a1n \sqrt[n]{a}=a^{\frac{1}{n}} B. The law of exponents for dividing powers with the same bases (in the opposite direction):

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

Let's start by changing the square roots to exponents using the law of exponents shown in A:

25=212512= \sqrt{2}\cdot\sqrt{5}= \\ \downarrow\\ 2^{\frac{1}{2}}\cdot5^{\frac{1}{2}}= We continue: since we are multiplying two terms with equal exponents we can use the law of exponents shown in B and combine them together as the same base raised to the same power:

212512=(25)12=1012=10 2^{\frac{1}{2}}\cdot5^{\frac{1}{2}}= \\ (2\cdot5)^{\frac{1}{2}}=\\ 10^{\frac{1}{2}}=\\ \boxed{\sqrt{10}} In the last steps wemultiplied the bases and then used the definition of the root as an exponent shown earlier in A (in the opposite direction) to return to the root notation.

Therefore, the correct answer is answer B.

Answer

10 \sqrt{10}