Simplifying Expressions with Square Roots and Surds

The problem asks us to simplify a given expression involving square roots and fractions, express it in the form \( a + b\sqrt 2 \), and then find the value of \( (3a + 4b) \).

The given expression is: \( \frac{5}{4{\sqrt 2 }} + \frac{{3 + 2\sqrt 2 }}{{3 - 2\sqrt 2 }} - \frac{{3 - 2\sqrt 2 }}{{3 + 2\sqrt 2 }} \)

Let's simplify each term separately.

Simplifying the First Term: \( \frac{5}{4{\sqrt 2 }} \)

To simplify this term, we need to rationalize the denominator. We multiply the numerator and denominator by \( \sqrt 2 \).

\( \frac{5}{4{\sqrt 2 }} = \frac{5}{4{\sqrt 2 }} \times \frac{\sqrt 2}{\sqrt 2} = \frac{5\sqrt 2}{4 \times (\sqrt 2)^2} = \frac{5\sqrt 2}{4 \times 2} = \frac{5\sqrt 2}{8} \)

Simplifying the Second Term: \( \frac{{3 + 2\sqrt 2 }}{{3 - 2\sqrt 2 }} \)

To simplify this term, we rationalize the denominator by multiplying the numerator and denominator by the conjugate of \( (3 - 2\sqrt 2) \), which is \( (3 + 2\sqrt 2) \).

\( \frac{{3 + 2\sqrt 2 }}{{3 - 2\sqrt 2 }} = \frac{{(3 + 2\sqrt 2)(3 + 2\sqrt 2) }}{{(3 - 2\sqrt 2)(3 + 2\sqrt 2) }} \)

Using the identities \( (x+y)^2 = x^2 + 2xy + y^2 \) and \( (x-y)(x+y) = x^2 - y^2 \):

• Numerator: \( (3 + 2\sqrt 2)^2 = 3^2 + 2(3)(2\sqrt 2) + (2\sqrt 2)^2 = 9 + 12\sqrt 2 + (4 \times 2) = 9 + 12\sqrt 2 + 8 = 17 + 12\sqrt 2 \)
• Denominator: \( (3 - 2\sqrt 2)(3 + 2\sqrt 2) = 3^2 - (2\sqrt 2)^2 = 9 - (4 \times 2) = 9 - 8 = 1 \)

So, \( \frac{{3 + 2\sqrt 2 }}{{3 - 2\sqrt 2 }} = \frac{17 + 12\sqrt 2}{1} = 17 + 12\sqrt 2 \)

Simplifying the Third Term: \( \frac{{3 - 2\sqrt 2 }}{{3 + 2\sqrt 2 }} \)

To simplify this term, we rationalize the denominator by multiplying the numerator and denominator by the conjugate of \( (3 + 2\sqrt 2) \), which is \( (3 - 2\sqrt 2) \).

\( \frac{{3 - 2\sqrt 2 }}{{3 + 2\sqrt 2 }} = \frac{{(3 - 2\sqrt 2)(3 - 2\sqrt 2) }}{{(3 + 2\sqrt 2)(3 - 2\sqrt 2) }} \)

Using the identities \( (x-y)^2 = x^2 - 2xy + y^2 \) and \( (x+y)(x-y) = x^2 - y^2 \):

• Numerator: \( (3 - 2\sqrt 2)^2 = 3^2 - 2(3)(2\sqrt 2) + (2\sqrt 2)^2 = 9 - 12\sqrt 2 + (4 \times 2) = 9 - 12\sqrt 2 + 8 = 17 - 12\sqrt 2 \)
• Denominator: \( (3 + 2\sqrt 2)(3 - 2\sqrt 2) = 3^2 - (2\sqrt 2)^2 = 9 - (4 \times 2) = 9 - 8 = 1 \)

So, \( \frac{{3 - 2\sqrt 2 }}{{3 + 2\sqrt 2 }} = \frac{17 - 12\sqrt 2}{1} = 17 - 12\sqrt 2 \)

Combining the Simplified Terms

Now, we combine the simplified terms according to the original expression:

\( \frac{5}{4{\sqrt 2 }} + \frac{{3 + 2\sqrt 2 }}{{3 - 2\sqrt 2 }} - \frac{{3 - 2\sqrt 2 }}{{3 + 2\sqrt 2 }} = \frac{5\sqrt 2}{8} + (17 + 12\sqrt 2) - (17 - 12\sqrt 2) \)

\( = \frac{5\sqrt 2}{8} + 17 + 12\sqrt 2 - 17 + 12\sqrt 2 \)

Combine the rational parts and the irrational parts:

• Rational parts: \( 17 - 17 = 0 \)
• Irrational parts: \( \frac{5\sqrt 2}{8} + 12\sqrt 2 + 12\sqrt 2 = \frac{5\sqrt 2}{8} + 24\sqrt 2 \)

Combine the irrational parts by finding a common denominator:

\( \frac{5\sqrt 2}{8} + 24\sqrt 2 = \frac{5\sqrt 2}{8} + \frac{24\sqrt 2 \times 8}{8} = \frac{5\sqrt 2 + 192\sqrt 2}{8} = \frac{(5 + 192)\sqrt 2}{8} = \frac{197\sqrt 2}{8} \)

So, the simplified expression is \( \frac{197}{8}\sqrt 2 \).

Equating to \( a + b\sqrt 2 \)

We are given that the expression equals \( a + b\sqrt 2 \).

\( \frac{197}{8}\sqrt 2 = a + b\sqrt 2 \)

Comparing the rational and irrational parts on both sides:

• The rational part on the left is 0. So, \( a = 0 \).
• The irrational part on the left is \( \frac{197}{8}\sqrt 2 \), which means the coefficient of \( \sqrt 2 \) is \( \frac{197}{8} \). So, \( b = \frac{197}{8} \).

Calculating \( (3a + 4b) \)

Now we substitute the values of \( a \) and \( b \) into the expression \( (3a + 4b) \).

\( 3a + 4b = 3(0) + 4\left(\frac{197}{8}\right) \)

\( = 0 + \frac{4 \times 197}{8} \)

We can cancel the 4 in the numerator and the 8 in the denominator:

\( = \frac{197}{2} \)

Converting the improper fraction to a mixed number:

\( \frac{197}{2} = \frac{196 + 1}{2} = \frac{196}{2} + \frac{1}{2} = 98 + \frac{1}{2} = 98\frac{1}{2} \)

Therefore, the value of \( (3a + 4b) \) is \( 98\frac{1}{2} \).

Revision Table - Simplifying Expressions

Additional Information - Working with Surds

Working with surds (irrational numbers involving roots) often requires rationalizing denominators and using algebraic identities.

• Rationalizing the Denominator: This process removes the surd from the denominator of a fraction, making it easier to work with. For a single term \( \frac{1}{\sqrt{x}} \), multiply by \( \frac{\sqrt{x}}{\sqrt{x}} \). For a binomial denominator like \( (p \pm q\sqrt{r}) \), multiply by its conjugate \( (p \mp q\sqrt{r}) \).
• Conjugates: The conjugate of \( (p + q\sqrt{r}) \) is \( (p - q\sqrt{r}) \), and vice versa. The product of a binomial involving a surd and its conjugate is always a rational number: \( (p + q\sqrt{r})(p - q\sqrt{r}) = p^2 - (q\sqrt{r})^2 = p^2 - q^2r \).
• Combining Like Surds: Just like algebraic terms (e.g., \( 3x + 5x = 8x \)), surds with the same root and radicand can be combined (e.g., \( 3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2} \)). Different surds (e.g., \( \sqrt{2} \) and \( \sqrt{3} \)) cannot be directly added or subtracted unless their radicands can be simplified to be the same.
• Algebraic Identities: Identities like \( (a+b)^2 \), \( (a-b)^2 \), and \( (a+b)(a-b) \) are frequently used when simplifying expressions involving surds, especially during rationalization.

Mastering these techniques is crucial for simplifying expressions containing square roots and other radicals in algebra.