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# Algebra

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Fully simplify $$\sqrt {49 - 20\sqrt {6}}$$

Jul 9, 2024

### 1+0 Answers

#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