If \(\frac {8 + 2\sqrt 3}{3\sqrt 3 + 5} = a\sqrt 3 - b,\) then the value of a + b is equal to:
18
Understanding the Problem: Rationalizing Surds and Finding a + b
The problem asks us to find the value of \(a + b\) given an equation where a fraction with surds in the denominator is equal to an expression of the form \(a\sqrt 3 - b\). To solve this, we first need to simplify the given fraction by rationalizing the denominator. Rationalizing the denominator helps us to rewrite the expression in a simpler form that can be easily compared to \(a\sqrt 3 - b\).
Step-by-Step Solution: Rationalizing the Denominator
The given equation is:
\(\frac {8 + 2\sqrt 3}{3\sqrt 3 + 5} = a\sqrt 3 - b\)
We will start by rationalizing the left side of the equation. The denominator is \(3\sqrt 3 + 5\). The conjugate of \(3\sqrt 3 + 5\) is \(3\sqrt 3 - 5\). We multiply both the numerator and the denominator by this conjugate.
LHS \( = \frac {8 + 2\sqrt 3}{3\sqrt 3 + 5} \times \frac {3\sqrt 3 - 5}{3\sqrt 3 - 5}\)
Multiplying the Numerator
Numerator \( = (8 + 2\sqrt 3)(3\sqrt 3 - 5)\)
We use the distributive property (or FOIL method) to multiply the terms:
- \(8 \times (3\sqrt 3) = 24\sqrt 3\)
- \(8 \times (-5) = -40\)
- \((2\sqrt 3) \times (3\sqrt 3) = 2 \times 3 \times \sqrt 3 \times \sqrt 3 = 6 \times 3 = 18\)
- \((2\sqrt 3) \times (-5) = -10\sqrt 3\)
Adding these terms:
Numerator \( = 24\sqrt 3 - 40 + 18 - 10\sqrt 3\)
Combine like terms (terms with \(\sqrt 3\) and constant terms):
Numerator \( = (24\sqrt 3 - 10\sqrt 3) + (-40 + 18)\)
Numerator \( = 14\sqrt 3 - 22\)
Multiplying the Denominator
Denominator \( = (3\sqrt 3 + 5)(3\sqrt 3 - 5)\)
This is in the form \((x+y)(x-y) = x^2 - y^2\), where \(x = 3\sqrt 3\) and \(y = 5\).
Denominator \( = (3\sqrt 3)^2 - 5^2\)
Calculate \((3\sqrt 3)^2\):
\((3\sqrt 3)^2 = 3^2 \times (\sqrt 3)^2 = 9 \times 3 = 27\)
Calculate \(5^2\):
\(5^2 = 25\)
Denominator \( = 27 - 25 = 2\)
Simplifying the Fraction
Now we have the simplified numerator and denominator:
LHS \( = \frac{14\sqrt 3 - 22}{2}\)
We can divide each term in the numerator by the denominator:
LHS \( = \frac{14\sqrt 3}{2} - \frac{22}{2}\)
LHS \( = 7\sqrt 3 - 11\)
Comparing and Finding a and b
Now we compare the simplified left side with the right side of the original equation:
\(7\sqrt 3 - 11 = a\sqrt 3 - b\)
By comparing the coefficients of \(\sqrt 3\) and the constant terms on both sides, we can find the values of \(a\) and \(b\).
- Coefficient of \(\sqrt 3\): \(a = 7\)
- Constant term: \(-b = -11\), which means \(b = 11\)
Calculating a + b
The problem asks for the value of \(a + b\).
\(a + b = 7 + 11\)
\(a + b = 18\)
Thus, the value of \(a + b\) is 18.
| Operation | Calculation |
|---|---|
| Rationalize Denominator | Multiply by \(\frac{3\sqrt 3 - 5}{3\sqrt 3 - 5}\) |
| Numerator Product | \((8 + 2\sqrt 3)(3\sqrt 3 - 5) = 14\sqrt 3 - 22\) |
| Denominator Product | \((3\sqrt 3 + 5)(3\sqrt 3 - 5) = 2\) |
| Simplified Expression | \(\frac{14\sqrt 3 - 22}{2} = 7\sqrt 3 - 11\) |
| Compare with \(a\sqrt 3 - b\) | \(a = 7\), \(b = 11\) |
| Calculate \(a+b\) | \(7 + 11 = 18\) |
Revision Table: Key Concepts
| Concept | Description |
|---|---|
| Rationalizing Denominator | Eliminating surds (irrational numbers) from the denominator of a fraction, typically by multiplying by the conjugate. |
| Conjugate of \(x\sqrt a + y\sqrt b\) | \(x\sqrt a - y\sqrt b\) (or \(x\sqrt a - y\) for \(x\sqrt a + y\)). The product of a binomial and its conjugate is always rational: \((x\sqrt a + y)(x\sqrt a - y) = (x\sqrt a)^2 - y^2\). |
| Comparing Expressions | If \(c\sqrt d + e = f\sqrt d + g\), then \(c=f\) and \(e=g\), provided \(\sqrt d\) is an irrational surd and \(c, d, e, f, g\) are rational numbers. |
Additional Information: Surds and Rational Numbers
Surds are irrational numbers that can be expressed as the root of a number, like \(\sqrt 2\), \(\sqrt 3\), \(\sqrt[3] 5\), etc. A number is a surd if its root cannot be simplified into a rational number. For example, \(\sqrt 4\) is not a surd because \(\sqrt 4 = 2\), which is a rational number.
A rational number is any number that can be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\). Examples include \(1\), \(-5\), \(1/2\), \(0.75\).
When we rationalize a denominator containing surds, we are essentially changing the form of the expression so that the denominator becomes a rational number, making it easier to work with or compare.
In this problem, after rationalizing, the expression \(7\sqrt 3 - 11\) is in the form \(a\sqrt 3 - b\), where \(a=7\) and \(b=11\). Both \(a\) and \(b\) are rational numbers (integers, specifically), which is typically expected in such problems unless otherwise stated.
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The product of \(\sqrt{2}\) and \(\sqrt{3}\) is:
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The decimal expansion of \(\frac{27}{25}\) will terminate after:
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- \((\sqrt2 -\sqrt3)^2\) is:
