Multiply Square Roots: √5 × √10 × √2 × √4 Step-by-Step Solution

Question

Solve the following exercise:

51024= \sqrt{5}\cdot\sqrt{10}\cdot\sqrt{2}\cdot\sqrt{4}=

Video Solution

Solution Steps

00:00 Simplify the following problem
00:03 When multiplying the root of a number (A) by the root of another number (B)
00:06 The result equals the root of their product (A times B)
00:09 Apply this formula to our exercise and calculate the multiplications
00:17 Calculate each multiplication separately
00:24 Break down 400 into 20 squared
00:27 The root of any squared number cancels out the square
00:30 This is the solution

Step-by-Step Solution

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

a. Root definition as an exponent:

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

b. The law of exponents for exponents applied to multiplication of terms in parentheses (in reverse order):

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:

51024=5121012212412= \sqrt{5}\cdot\sqrt{10}\cdot\sqrt{2}\cdot\sqrt{4}= \\ \downarrow\\ 5^{\frac{1}{2}}\cdot10^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\cdot4^{\frac{1}{2}}=

Due to the fact that there is a multiplication operation between four terms with identical exponents we are able to apply the law of exponents mentioned in b (which also applies to multiplication of several terms in parentheses) Combine them together in a multiplication operation inside of parentheses that are also raised to the same exponent:

5121012212412=(51024)12=40012=400=20 5^{\frac{1}{2}}\cdot10^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\cdot4^{\frac{1}{2}}=\\ (5\cdot10\cdot2\cdot4)^{\frac{1}{2}}=\\ 400^{\frac{1}{2}}=\\ \sqrt{400}=\\ \boxed{20}

In the final stages, we first performed the multiplication within the parentheses, then we once again used the root definition as an exponent mentioned earlier in a (in reverse order) to return to root notation, and in the final stage we calculated the known square root of 400.

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

Answer

20 20