If \(x = \sqrt {-\sqrt 3 + \sqrt {3 + 8\sqrt {7 + 4\sqrt 3}}}\) where x > 0, then the value of x is equal to:
2
Simplifying Nested Square Roots to Find x
The problem asks us to find the value of \(x\), where \(x > 0\), given a complex expression involving nested square roots. To solve this, we need to simplify the expression step-by-step, starting from the innermost square root.
The given expression is:
\(x = \sqrt {-\sqrt 3 + \sqrt {3 + 8\sqrt {7 + 4\sqrt 3}}}\)
Step 1: Simplify the Innermost Square Root
The innermost square root is \( \sqrt{7 + 4\sqrt 3} \). We look for two numbers whose sum of squares is 7 and whose product is \(2\sqrt{3}\) (since \(4\sqrt{3} = 2 \times 2\sqrt{3}\)). We can try to write \(7 + 4\sqrt{3}\) in the form \((a+b\sqrt{c})^2\) or \((a+b)^2\).
Consider the form \((a+b)^2 = a^2 + 2ab + b^2\). We want \(a^2 + b^2 = 7\) and \(2ab = 4\sqrt{3}\), which means \(ab = 2\sqrt{3}\). If we consider \(a=2\) and \(b=\sqrt{3}\), then \(a^2 = 4\) and \(b^2 = 3\). Their sum is \(4+3=7\). Their product \(ab = 2\sqrt{3}\), so \(2ab = 4\sqrt{3}\).
Thus, \(7 + 4\sqrt 3 = (2)^2 + 2(2)(\sqrt 3) + (\sqrt 3)^2 = (2 + \sqrt 3)^2\).
So, \( \sqrt{7 + 4\sqrt 3} = \sqrt{(2 + \sqrt 3)^2} \).
Since \(2 + \sqrt 3\) is positive, \( \sqrt{(2 + \sqrt 3)^2} = 2 + \sqrt 3 \).
Step 2: Substitute and Simplify the Next Layer
Now substitute the simplified part back into the main expression:
\(x = \sqrt {-\sqrt 3 + \sqrt {3 + 8(2 + \sqrt 3)}}\)
Simplify the expression inside the inner square root:
\(3 + 8(2 + \sqrt 3) = 3 + 16 + 8\sqrt 3 = 19 + 8\sqrt 3\)
The expression becomes:
\(x = \sqrt {-\sqrt 3 + \sqrt {19 + 8\sqrt 3}}\)
Step 3: Simplify the New Inner Square Root
Now we need to simplify \( \sqrt{19 + 8\sqrt 3} \). We look for two numbers whose sum of squares is 19 and whose product is \(4\sqrt{3}\) (since \(8\sqrt{3} = 2 \times 4\sqrt{3}\)).
Consider the form \((a+b)^2 = a^2 + 2ab + b^2\). We want \(a^2 + b^2 = 19\) and \(2ab = 8\sqrt{3}\), which means \(ab = 4\sqrt{3}\). If we consider \(a=4\) and \(b=\sqrt{3}\), then \(a^2 = 16\) and \(b^2 = 3\). Their sum is \(16+3=19\). Their product \(ab = 4\sqrt{3}\), so \(2ab = 8\sqrt{3}\).
Thus, \(19 + 8\sqrt 3 = (4)^2 + 2(4)(\sqrt 3) + (\sqrt 3)^2 = (4 + \sqrt 3)^2\).
So, \( \sqrt{19 + 8\sqrt 3} = \sqrt{(4 + \sqrt 3)^2} \).
Since \(4 + \sqrt 3\) is positive, \( \sqrt{(4 + \sqrt 3)^2} = 4 + \sqrt 3 \).
Step 4: Substitute and Find the Value of x
Substitute this result back into the main expression:
\(x = \sqrt {-\sqrt 3 + (4 + \sqrt 3)}\)
Simplify the expression inside the remaining square root:
\(-\sqrt 3 + 4 + \sqrt 3 = 4\)
So, the expression simplifies to:
\(x = \sqrt {4}\)
Since the problem states that \(x > 0\), we take the positive square root.
\(x = 2\)
The value of \(x\) is 2.
Conclusion
By simplifying the nested square roots step-by-step, we found that the value of \(x\) is 2.
The value 2 corresponds to option 2.
Revision Table: Key Concepts for Simplifying Radicals
| Concept | Description | Formula Example |
|---|---|---|
| Simplifying \( \sqrt{(a+b)^2} \) | If \(a+b \ge 0\), then \( \sqrt{(a+b)^2} = a+b \). This is crucial for simplifying expressions like \( \sqrt{A + B\sqrt{C}} \). | \( \sqrt{(3+2)^2} = \sqrt{5^2} = 5 \) |
| Simplifying \( \sqrt{a + b\sqrt{c}} \) | Try to write \(a + b\sqrt{c}\) as a perfect square like \( (d+e\sqrt{c})^2 \) or \( (d+e)^2 \) if \(c\) is 1. Look for terms \(d, e\) such that \(d^2 + e^2 = a\) and \(2de = b\) (if \(c=1\)) or \(d^2 + e^2c = a\) and \(2de = b\) (if it's in the form \( \sqrt{(d+e\sqrt{c})^2} \)). | \( \sqrt{5 + 2\sqrt{6}} = \sqrt{(\sqrt{3}+\sqrt{2})^2} = \sqrt{3}+\sqrt{2} \) because \( (\sqrt{3})^2 + (\sqrt{2})^2 = 3+2=5 \) and \( 2(\sqrt{3})(\sqrt{2}) = 2\sqrt{6} \). |
| Order of Operations | Always simplify from the innermost part of the expression outwards, especially with nested roots or parentheses. | Evaluate \( \sqrt{5 + \sqrt{16}} \) by first evaluating \( \sqrt{16} = 4 \), then \( \sqrt{5+4} = \sqrt{9} = 3 \). |
Additional Information: Understanding Perfect Squares under Square Roots
The key technique used in this problem is recognizing when an expression under a square root is a perfect square. Specifically, expressions in the form \( a + b\sqrt{c} \) can sometimes be written as \( (d + e\sqrt{c})^2 \) or \( (d+e)^2 \) if \(c=1\).
For \( a + b\sqrt{c} \), we want to find \(d\) and \(e\) such that:
\((d + e\sqrt{c})^2 = d^2 + 2de\sqrt{c} + (e\sqrt{c})^2 = (d^2 + e^2c) + (2de)\sqrt{c}\)
Comparing this to \( a + b\sqrt{c} \), we need to solve the system of equations:
\(d^2 + e^2c = a\)
\(2de = b\)
In our problem, for \( 7 + 4\sqrt 3 \), we had \(a=7, b=4, c=3\). We looked for \(d, e\) such that \(d^2 + e^2 \times 3 = 7\) and \(2de = 4\). The second equation gives \(de=2\). Trying integer/simple values for \(d, e\), if \(d=2, e=1\), then \(d^2+e^2 \times 3 = 2^2 + 1^2 \times 3 = 4 + 3 = 7\). This works! So \(7+4\sqrt{3} = (2+1\sqrt{3})^2 = (2+\sqrt{3})^2\).
For \( 19 + 8\sqrt 3 \), we had \(a=19, b=8, c=3\). We looked for \(d, e\) such that \(d^2 + e^2 \times 3 = 19\) and \(2de = 8\). The second equation gives \(de=4\). Trying integer/simple values for \(d, e\), if \(d=4, e=1\), then \(d^2+e^2 \times 3 = 4^2 + 1^2 \times 3 = 16 + 3 = 19\). This works! So \(19+8\sqrt{3} = (4+1\sqrt{3})^2 = (4+\sqrt{3})^2\).
Recognizing this pattern is key to simplifying complex nested radical expressions like the one in this problem.
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