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Fully simplify \(\sqrt {49 - 20\sqrt {6}}\)

 Jul 9, 2024
 #1
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Let's break down the expression step by step.

 

sqrt(49 - 20*sqrt(6)) = ?

 

First, we can start by evaluating the expression inside the parentheses:

 

49 - 20*sqrt(6) = ?

 

We can use the distributive property to multiply 20 by sqrt(6):

 

49 - 20*sqrt(6) = 49 - 20(sqrt(6))

 

Now, we can simplify this expression by recognizing that 49 is a perfect square:

 

49 = 7^2

 

So, we can rewrite the expression as:

 

49 - 20(sqrt(6)) = 7^2 - 20(sqrt(6))

 

Next, we can use the difference of squares formula to simplify further:

 

a^2 - b^2 = (a + b)(a - b)

 

In this case, a = 7 and b = sqrt(6), so we get:

 

7^2 - sqrt(6)^2 = (7 + sqrt(6))(7 - sqrt(6))

 

Now, we can simplify the expression by combining like terms:

 

(7 + sqrt(6))(7 - sqrt(6)) = ?

 

To combine these terms, we can use the fact that sqrt(x)^2 = x:

 

7 + sqrt(6) and 7 - sqrt(6) are both factors of (7)^2 - (sqrt(6))^2

 

So, we can factor out a common factor of (7 - sqrt(6)):

 

= (7 - sqrt(6))(7 + sqrt(6))

 

And that's the simplified form of the original expression!

 

sqrt(49 - 20*sqrt(6)) = 7 - sqrt(6)

 Jul 9, 2024

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