If \(a\; = \;\sqrt {7\; + \;4\sqrt 3 ,}\) then what is the value of a + 1/a?
4
Finding the Value of a + 1/a Given a Square Root Expression
We are given the value of \(a\) as a square root expression: \(a\; = \;\sqrt {7\; + \;4\sqrt 3 }\). We need to find the value of \(a + 1/a\).
Simplifying the Expression for 'a'
First, let's simplify the expression under the square root sign. The term \(7 + 4\sqrt 3\) looks like it could be a perfect square of the form \((x+y)^2\). Recall that \((x+y)^2 = x^2 + 2xy + y^2\).
We want to express \(7 + 4\sqrt 3\) in the form \(x^2 + 2xy + y^2\). Comparing the terms:
- The term \(4\sqrt 3\) corresponds to \(2xy\). So, \(2xy = 4\sqrt 3\), which means \(xy = 2\sqrt 3\).
- The constant term \(7\) corresponds to \(x^2 + y^2\). So, \(x^2 + y^2 = 7\).
We need to find two numbers, \(x\) and \(y\), whose product is \(2\sqrt 3\) and the sum of whose squares is \(7\). Let's consider pairs of factors for \(2\sqrt 3\). A possible pair is \(2\) and \(\sqrt 3\). Let's check if these values satisfy the second condition:
- Let \(x=2\) and \(y=\sqrt 3\).
- \(x^2 + y^2 = 2^2 + (\sqrt 3)^2 = 4 + 3 = 7\). This matches the constant term.
Thus, \(7 + 4\sqrt 3\) can be written as \((2 + \sqrt 3)^2\).
Now, substitute this back into the expression for \(a\):
\(a = \sqrt{7 + 4\sqrt 3} = \sqrt{(2 + \sqrt 3)^2}\)
Since \((2 + \sqrt 3)\) is a positive value, \(\sqrt{(2 + \sqrt 3)^2} = 2 + \sqrt 3\).
So, \(a = 2 + \sqrt 3\).
Calculating 1/a
Next, we need to find the value of \(1/a\):
\(1/a = 1 / (2 + \sqrt 3)\)
To simplify this expression, we rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator, which is \(2 - \sqrt 3\).
\(1/a = \frac{1}{2 + \sqrt 3} \times \frac{2 - \sqrt 3}{2 - \sqrt 3}\)
Using the difference of squares formula, \((p+q)(p-q) = p^2 - q^2\), the denominator becomes:
\((2 + \sqrt 3)(2 - \sqrt 3) = 2^2 - (\sqrt 3)^2 = 4 - 3 = 1\)
So, the expression for \(1/a\) simplifies to:
\(1/a = \frac{2 - \sqrt 3}{1} = 2 - \sqrt 3\)
Finding the Value of a + 1/a
Now we can find the value of \(a + 1/a\) by substituting the simplified values of \(a\) and \(1/a\):
\(a + 1/a = (2 + \sqrt 3) + (2 - \sqrt 3)\)
\(a + 1/a = 2 + \sqrt 3 + 2 - \sqrt 3\)
Combine the like terms:
\(a + 1/a = (2 + 2) + (\sqrt 3 - \sqrt 3)\)
\(a + 1/a = 4 + 0\)
\(a + 1/a = 4\)
The value of \(a + 1/a\) is 4.
Let's check this against the given options:
| Option | Value |
|---|---|
| 1 | 2 |
| 2 | 3 |
| 3 | 4 |
| 4 | 7 |
Our calculated value, 4, matches Option 3.
Revision Table: Simplifying Square Roots and Reciprocals
| Concept | Method | Example |
|---|---|---|
| Simplifying \(\sqrt{x+y\sqrt{z}}\) | Try to express \(x+y\sqrt{z}\) as a perfect square \((a+b\sqrt{z'})^2\) or \((a\sqrt{k}+b\sqrt{m})^2\). Look for \(2ab\sqrt{z'}\) term matching \(y\sqrt{z}\) and \(a^2+b^2z'\) matching \(x\). | \(\sqrt{7+4\sqrt{3}} = \sqrt{(2+\sqrt{3})^2} = 2+\sqrt{3}\) |
| Rationalizing \(1/(p+q\sqrt{r})\) | Multiply numerator and denominator by the conjugate \(p-q\sqrt{r}\). Use \((p+q\sqrt{r})(p-q\sqrt{r}) = p^2 - q^2r\). | \(1/(2+\sqrt{3}) = (2-\sqrt{3})/((2+\sqrt{3})(2-\sqrt{3})) = (2-\sqrt{3})/(4-3) = 2-\sqrt{3}\) |
| Finding \(a+1/a\) for \(a=p+\sqrt{q}\) where \(p^2-q=1\) | If \(a=p+\sqrt{q}\) and \(1/a=p-\sqrt{q}\) (because \(p^2-q=1\)), then \(a+1/a = (p+\sqrt{q}) + (p-\sqrt{q}) = 2p\). | For \(a=2+\sqrt{3}\), \(p=2, q=3\). \(p^2-q = 2^2-3 = 4-3=1\). So \(1/a=2-\sqrt{3}\) and \(a+1/a = 2 \times 2 = 4\). |
Additional Information on Surds and Rationalization
Surds are irrational numbers that are expressed using a root symbol (\(\sqrt{}, \sqrt[3]{}\), etc.). Examples include \(\sqrt 2\), \(\sqrt 3\), \(2\sqrt 5\).
Rationalization is the process of converting an expression with an irrational denominator into an equivalent expression with a rational denominator. This is often done by multiplying the numerator and denominator by the conjugate of the denominator.
The conjugate of an expression like \(p + q\sqrt r\) is \(p - q\sqrt r\). When you multiply an expression by its conjugate, you get a rational number (if \(r\) is rational) because of the difference of squares property: \((p + q\sqrt r)(p - q\sqrt r) = p^2 - (q\sqrt r)^2 = p^2 - q^2r\).
In this problem, \(a = 2 + \sqrt 3\). Its conjugate is \(2 - \sqrt 3\). Notice that \(a \times (2 - \sqrt 3) = (2 + \sqrt 3)(2 - \sqrt 3) = 2^2 - (\sqrt 3)^2 = 4 - 3 = 1\). This means \(2 - \sqrt 3 = 1/a\). This is a common pattern when dealing with numbers of the form \(p + \sqrt q\) where \(p^2 - q = 1\).
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